-
Notifications
You must be signed in to change notification settings - Fork 2
/
s06-04-solving-linear-systems-with-th.html
937 lines (920 loc) · 144 KB
/
s06-04-solving-linear-systems-with-th.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<link href="shared/bookhub.css" rel="stylesheet" type="text/css">
<title>Solving Linear Systems with Three Variables</title>
</head>
<body>
<div id=navbar-top class="navbar">
<div class="navbar-part left">
<a href="s06-03-applications-of-linear-systems.html"><img src="shared/images/batch-left.png"></a> <a href="s06-03-applications-of-linear-systems.html">Previous Section</a>
</div>
<div class="navbar-part middle">
<a href="index.html"><img src="shared/images/batch-up.png"></a> <a href="index.html">Table of Contents</a>
</div>
<div class="navbar-part right">
<a href="s06-05-matrices-and-gaussian-eliminat.html">Next Section</a> <a href="s06-05-matrices-and-gaussian-eliminat.html"><img src="shared/images/batch-right.png"></a>
</div>
</div>
<div id="book-content">
<div class="section" id="fwk-redden-ch03_s04" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">3.4</span> Solving Linear Systems with Three Variables</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch03_s04_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch03_s04_o01" numeration="arabic">
<li>Check solutions to linear systems with three variables.</li>
<li>Solve linear systems with three variables by elimination.</li>
<li>Identify dependent and inconsistent systems.</li>
<li>Solve applications involving three unknowns.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch03_s04_s01" version="5.0" lang="en">
<h2 class="title editable block">Solutions to Linear Systems with Three Variables</h2>
<p class="para editable block" id="fwk-redden-ch03_s04_s01_p01">Real-world applications are often modeled using more than one variable and more than one equation. In this section, we will study linear systems consisting of three linear equations each with three variables. For example,</p>
<p class="para block" id="fwk-redden-ch03_s04_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0448" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mtext> </mtext><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd><mtd><mspace width="2em"></mspace><mstyle color="#007fbf"><mi>(</mi><mi>1</mi><mi>)</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>x</mi><mo>−</mo><mtext> </mtext><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd><mtd><mspace width="2em"></mspace><mstyle color="#007fbf"><mi>(</mi><mi>2</mi><mi>)</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd><mspace width="2em"></mspace><mstyle color="#007fbf"><mi>(</mi><mi>3</mi><mi>)</mi></mstyle></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch03_s04_s01_p03">A solution to such a linear system is an <span class="margin_term"><a class="glossterm">ordered triple</a><span class="glossdef">Triples (<em class="emphasis">x</em>, <em class="emphasis">y</em>, <em class="emphasis">z</em>) that identify position relative to the origin in three-dimensional space.</span></span> (<em class="emphasis">x</em>, <em class="emphasis">y</em>, <em class="emphasis">z</em>) that solves all of the equations. In this case, (−2, 1, 3) is the only solution. To check that an ordered triple is a solution, substitute in the corresponding <em class="emphasis">x</em>-, <em class="emphasis">y</em>-, and <em class="emphasis">z</em>-values and then simplify to see if you obtain a true statement from all three equations.</p>
<div class="informaltable block">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0449" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi><mi>:</mi><mtext> </mtext></mstyle><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></math></span></p></th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0450" display="inline"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mi>E</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mstyle color="#007fbf"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>:</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo stretchy="false">)</mo></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mo>+</mo><mn>2</mn><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0451" display="inline"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mi>E</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mstyle color="#007fbf"><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>:</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>3</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo stretchy="false">)</mo></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>12</mn><mo>−</mo><mn>1</mn><mo>−</mo><mn>9</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>4</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0452" display="inline"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mi>E</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mstyle color="#007fbf"><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>:</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mo stretchy="false">(</mo><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle><mo stretchy="false">)</mo><mo>+</mo><mn>10</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo stretchy="false">)</mo></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>2</mn><mo>+</mo><mn>10</mn><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch03_s04_s01_p06">Because the ordered triple satisfies all three equations we conclude that it is indeed a solution.</p>
<div class="callout block" id="fwk-redden-ch03-s04_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch03_s04_s01_p07">Determine whether or not <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0453" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mn>4</mn><mo>,</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span> is a solution to the following linear system:</p>
<p class="para" id="fwk-redden-ch03_s04_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0454" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>9</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>6</mn><mi>z</mi><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>7</mn><mi>z</mi><mo>=</mo><mn>15</mn></mtd></mtr></mtable></mrow><mtext> </mtext><mo>.</mo></mrow></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s04_s01_p09"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0455" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi><mtext>:</mtext><mtext> </mtext></mstyle><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>,</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>)</mo></mrow></math></span></p></th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0456" display="inline"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mi>E</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mstyle color="#007fbf"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>:</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>6</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>4</mn></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>6</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mo>+</mo><mn>4</mn><mo>−</mo><mn>8</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0457" display="inline"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mi>E</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mstyle color="#007fbf"><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>:</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>3</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>4</mn></mstyle><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>4</mn><mn>3</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mo>−</mo><mn>12</mn><mo>+</mo><mn>4</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>14</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>14</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0458" display="inline"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mi>E</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mstyle color="#007fbf"><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>:</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>7</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>4</mn></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>7</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>4</mn><mn>3</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mo>+</mo><mn>8</mn><mo>−</mo><mfrac><mrow><mn>28</mn></mrow><mn>3</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>5</mn><mn>3</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>15</mn><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch03_s04_s01_p10">Answer: The point does not satisfy all of the equations and thus is not a solution.</p>
</div>
<p class="para block" id="fwk-redden-ch03_s04_s01_p11">An ordered triple such as <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0459" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span> can be graphed in three-dimensional space as follows:</p>
<div class="informalfigure large block">
<img src="section_06/926cca2c2942c813865579781eab4bdf.png">
</div>
<p class="para block" id="fwk-redden-ch03_s04_s01_p13">The ordered triple indicates position relative to the origin (0, 0, 0), in this case, 2 units along the <em class="emphasis">x</em>-axis, 4 units parallel to the <em class="emphasis">y</em>-axis, and 5 units parallel to the <em class="emphasis">z</em>-axis. A <span class="margin_term"><a class="glossterm">linear equation with three variables</a><span class="glossdef">An equation that can be written in the standard form <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0460" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi><mi>z</mi><mo>=</mo><mi>d</mi></mrow></math></span> where <em class="emphasis">a</em>, <em class="emphasis">b</em>, <em class="emphasis">c</em>, and <em class="emphasis">d</em> are real numbers.</span></span> is in standard form if
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0461" display="block"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi><mi>z</mi><mo>=</mo><mi>d</mi></mrow></math></span>
where <em class="emphasis">a</em>, <em class="emphasis">b</em>, <em class="emphasis">c</em>, and <em class="emphasis">d</em> are real numbers. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0462" display="inline"><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>26</mn></mrow></math></span> is in standard form. Solving for <em class="emphasis">z</em>, we obtain <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0463" display="inline"><mrow><mi>z</mi><mo>=</mo><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>+</mo><mn>13</mn></mrow></math></span> and can consider both <em class="emphasis">x</em> and <em class="emphasis">y</em> to be the independent variables. When graphed in three-dimensional space, its graph will form a straight flat surface called a <span class="margin_term"><a class="glossterm">plane</a><span class="glossdef">Any flat two-dimensional surface.</span></span>.</p>
<div class="informalfigure large block">
<img src="section_06/cb50e8c9a978a4ca62a8c6d4a30d4c65.png">
</div>
<p class="para editable block" id="fwk-redden-ch03_s04_s01_p17">Therefore, the graph of a system of three linear equations and three unknowns will consist of three planes in space. If there is a simultaneous solution, the system is consistent and the solution corresponds to a point where the three planes intersect.</p>
<div class="informalfigure large block">
<img src="section_06/7459f0f0ab8b4fef9fb707a49ccbc0ae.png">
</div>
<p class="para editable block" id="fwk-redden-ch03_s04_s01_p19">Graphing planes in three-dimensional space is not within the scope of this textbook. However, it is always important to understand the geometric interpretation.</p>
<div class="callout block" id="fwk-redden-ch03-s04_s01_n01a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch03_s04_s01_p20"><strong class="emphasis bold">Try this!</strong> Determine whether or not (3, −1, 2) a solution to the system: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0464" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mi> </mi><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>17</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s04_s01_p21">Answer: Yes, it is a solution.</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/2UET4LzXoYg" condition="http://img.youtube.com/vi/2UET4LzXoYg/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/2UET4LzXoYg" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch03_s04_s02" version="5.0" lang="en">
<h2 class="title editable block">Solve Linear Systems with Three Variables by Elimination</h2>
<p class="para editable block" id="fwk-redden-ch03_s04_s02_p01">In this section, the elimination method is used to solve systems of three linear equations with three variables. The idea is to eliminate one of the variables and resolve the original system into a system of two linear equations, after which we can then solve as usual. The steps are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch03-s04_s02_n01">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch03_s04_s02_p02">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0465" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn><mtext> </mtext><mstyle color="#007fbf"><mtext fontstyle="italic">(1)</mtext></mstyle></mtd></mtr><mtr><mtd><mn>6</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn><mtext> </mtext><mstyle color="#007fbf"><mtext fontstyle="italic">(2)</mtext></mstyle><mtext> </mtext><mtext>.</mtext></mtd></mtr><mtr><mtd><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mtext> </mtext><mstyle color="#007fbf"><mtext fontstyle="italic">(3)</mtext></mstyle></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p03">All three equations are in standard form. If this were not the case, it would be a best practice to rewrite the equations in standard form before beginning this process.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p04"><strong class="emphasis bold">Step 1</strong>: Choose any two of the equations and eliminate a variable. In this case, we can line up the variable <em class="emphasis">z</em> to eliminate if we group 3 times the first equation with the second equation.</p>
<div class="informalfigure large">
<img src="section_06/516fe5445b196be3005983298be1a411.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s02_p06">Next, add the equations together.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p07"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0466" display="block"><mtable columnspacing="0.1em" rowlines="none solid none"><mtr><mtd></mtd><mtd columnalign="right"><mn>9</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mstyle color="#ff0000"><mtext> </mtext><mo>−</mo><mn>3</mn><mi>z</mi></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>21</mn></mtd><mtd></mtd></mtr><mtr><mtd style="border-bottom:1pt solid black"><mo>+</mo></mtd><mtd columnalign="right" style="border-bottom:1pt solid black"><mn>6</mn><mi>x</mi><mo>−</mo><mi>y</mi><mstyle color="#ff0000"><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>3</mn><mi>z</mi></mstyle></mtd><mtd style="border-bottom:1pt solid black"><mo>=</mo></mtd><mtd columnalign="left" style="border-bottom:1pt solid black"><mo>−</mo><mn>4</mn></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd columnalign="left"><mn>15</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd><mo>−</mo><mn>25</mn></mtd><mtd><mspace width="1em"></mspace><mstyle color="#007f3f"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p08"><strong class="emphasis bold">Step 2</strong>: Choose any other two equations and eliminate the same variable. We can line up <em class="emphasis">z</em> to eliminate again if we group −2 times the first equation with the third equation.</p>
<div class="informalfigure large">
<img src="section_06/9ebe0a82e20ab5ef2e179afa669cdc14.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s02_p10">And then add,</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p11"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0467" display="block"><mtable columnspacing="0.1em" rowlines="none solid none"><mtr><mtd></mtd><mtd columnalign="right"><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mstyle color="#ff0000"><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>2</mn><mi>z</mi></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd><mtd></mtd></mtr><mtr><mtd style="border-bottom:1pt solid black"><mo>+</mo></mtd><mtd columnalign="right" style="border-bottom:1pt solid black"><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mstyle color="#ff0000"><mo>−</mo><mn>2</mn><mi>z</mi></mstyle></mtd><mtd style="border-bottom:1pt solid black"><mo>=</mo></mtd><mtd columnalign="left" style="border-bottom:1pt solid black"><mn>2</mn></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>16</mn></mtd><mtd><mspace width="1em"></mspace><mstyle color="#007f3f"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p12"><strong class="emphasis bold">Step 3</strong>: Solve the resulting system of two equations with two unknowns. Here we solve by elimination. Multiply the second equation by 3 to line up the variable <em class="emphasis">x</em> to eliminate.</p>
<div class="informalfigure large">
<img src="section_06/479855f616ff4062df8e26f9d8ad2d39.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s02_p14">Next, add the equations together.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0468" display="block"><mtable columnspacing="0.1em" rowlines="none solid none"><mtr><mtd></mtd><mtd columnalign="right"><mstyle color="#ff0000"><mn>15</mn><mi>x</mi></mstyle><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>25</mn></mtd></mtr><mtr><mtd style="border-bottom:1pt solid black"><mo>+</mo></mtd><mtd style="border-bottom:1pt solid black" columnalign="right"><mstyle color="#ff0000"><mo>−</mo><mn>15</mn><mi>x</mi></mstyle><mo>+</mo><mn>18</mn><mi>y</mi></mtd><mtd style="border-bottom:1pt solid black"><mo>=</mo></mtd><mtd style="border-bottom:1pt solid black" columnalign="left"><mn>48</mn></mtd></mtr><mtr><mtd></mtd><mtd columnalign="right"><mn>23</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>23</mn></mtd></mtr><mtr><mtd></mtd><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p16"><strong class="emphasis bold">Step 4</strong>: Back substitute and determine all of the coordinates. To find <em class="emphasis">x</em> use the following,</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p17"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0469" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>15</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>25</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>15</mn><mi>x</mi><mo>+</mo><mn>5</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>25</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>15</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>30</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p18">Now choose one of the original equations to find <em class="emphasis">z</em>,</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p19"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0470" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn><mtext> </mtext><mstyle color="#007fbf"><mi>(</mi><mi mathvariant="italic">1</mi><mi>)</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mo>+</mo><mn>2</mn><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>4</mn><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p20">Hence the solution, presented as an ordered triple (<em class="emphasis">x</em>, <em class="emphasis">y</em>, <em class="emphasis">z</em>), is (−2, 1, 3). This is the same system that we checked in the beginning of this section.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p21">Answer: (−2, 1, 3)</p>
</div>
<p class="para editable block" id="fwk-redden-ch03_s04_s02_p22">It does not matter which variable we initially choose to eliminate, as long as we eliminate it twice with two different sets of equations.</p>
<div class="callout block" id="fwk-redden-ch03-s04_s02_n02">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch03_s04_s02_p23">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0471" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd><mn>9</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>6</mn><mi>z</mi><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>7</mn><mi>z</mi><mo>=</mo><mn>15</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p24">Because <em class="emphasis">y</em> has coefficient 1 in the second equation, choose to eliminate this variable. Use equations 1 and 2 to eliminate <em class="emphasis">y</em>.</p>
<div class="informalfigure large">
<img src="section_06/a930dc2e59fe26d4e9f63d71255af376.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s02_p26">Next use equations 2 and 3 to eliminate <em class="emphasis">y</em> again.</p>
<div class="informalfigure large">
<img src="section_06/1851087164f84417d08d2fca715fe95b.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s02_p28">This leaves a system of two equations with two variables <em class="emphasis">x</em> and <em class="emphasis">z</em>,</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p29"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0472" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>21</mn><mi>x</mi><mo>−</mo><mn>15</mn><mi>z</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>15</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p30">Multiply the second equation by 3 and eliminate the variable <em class="emphasis">z</em>.</p>
<div class="informalfigure large">
<img src="section_06/8f6730530de1044a573e39ba7e363c32.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s02_p32">Now back substitute to find <em class="emphasis">z</em>.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p33"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0473" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>21</mn><mi>x</mi><mo>−</mo><mn>15</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>21</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>15</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>14</mn><mo>−</mo><mn>15</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>15</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p34">Finally, choose one of the original equations to find <em class="emphasis">y</em>.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p35"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0474" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mo>−</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>1</mn><mo>−</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p36">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0475" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mn>5</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch03-s04_s02_n03">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch03_s04_s02_p37">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0476" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>7</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>12</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>13</mn></mtd></mtr></mtable></mrow><mo>.</mo></mrow></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p38">In this example, there is no obvious choice of variable to eliminate. We choose to eliminate <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p39"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0477" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd><mtable columnspacing="0.1em"><mtr><mtd><mrow><mstyle color="#007fbf"><mi>(</mi><mi>1</mi><mi>)</mi></mstyle></mrow></mtd></mtr><mtr><mtd><mrow><mstyle color="#007fbf"><mi>(</mi><mi>2</mi><mi>)</mi></mstyle></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mspace width="0.2em"></mspace><mn>2</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>7</mn><mi>z</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>=</mo><mn>12</mn></mtd></mtr></mtable></mrow></mtd><mtd><mtable columnspacing="0.1em"><mtr><mtd><mrow><mover><mo>⇒</mo><mrow><mo>×</mo><mn>3</mn></mrow></mover></mrow></mtd></mtr><mtr><mtd><mrow><munder><mo>⇒</mo><mrow><mo>×</mo><mn>2</mn></mrow></munder></mrow></mtd></mtr></mtable></mtd><mtd><munder accentunder="true"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>6</mn><mi>x</mi><mo>+</mo><mn>18</mn><mi>y</mi><mo>+</mo><mn>21</mn><mi>z</mi><mo>=</mo><mn>12</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>10</mn><mi>z</mi><mo>=</mo><mn>24</mn></mtd></mtr></mtable></mrow></mrow><mo stretchy="true">–</mo></munder></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd></mtd><mtd></mtd><mtd columnalign="right"><mrow><mn>10</mn><mi>y</mi><mo>+</mo><mn>31</mn><mi>z</mi><mo>=</mo><mn>36</mn></mrow></mtd><mtd><mrow><mstyle color="#007f3f"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p40">Next use equations 2 and 3 to eliminate <em class="emphasis">x</em> again.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p41"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0478" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd><mtable columnspacing="0.1em"><mtr><mtd><mrow><mstyle color="#007fbf"><mi>(</mi><mi>2</mi><mi>)</mi></mstyle></mrow></mtd></mtr><mtr><mtd><mrow><mstyle color="#007fbf"><mi>(</mi><mi>3</mi><mi>)</mi></mstyle></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>=</mo><mn>12</mn></mtd></mtr><mtr><mtd><mspace width="0.5em"></mspace><mn>5</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>13</mn></mtd></mtr></mtable></mrow></mtd><mtd><mtable columnspacing="0.1em"><mtr><mtd><mrow><mover><mo>⇒</mo><mrow><mo>×</mo><mn>5</mn></mrow></mover></mrow></mtd></mtr><mtr><mtd><mrow><munder><mo>⇒</mo><mrow><mo>×</mo><mn>3</mn></mrow></munder></mrow></mtd></mtr></mtable></mtd><mtd><mrow><munder accentunder="true"><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mo>−</mo><mn>15</mn><mi>x</mi><mo>−</mo><mn>20</mn><mi>y</mi><mo>+</mo><mn>25</mn><mi>z</mi><mo>=</mo><mn>60</mn></mtd></mtr><mtr><mtd><mspace width="0.7em"></mspace><mn>15</mn><mi>x</mi><mo>+</mo><mn>30</mn><mi>y</mi><mo>−</mo><mspace width="0.3em"></mspace><mn>9</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>39</mn></mtd></mtr></mtable></mrow><mo stretchy="true">–</mo></munder></mrow></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd></mtd><mtd></mtd><mtd columnalign="right"><mn>10</mn><mi>y</mi><mo>+</mo><mn>16</mn><mi>z</mi><mo>=</mo><mn>21</mn></mtd><mtd><mstyle color="#007f3f"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p42">This leaves a system of two equations with two variables <em class="emphasis">y</em> and <em class="emphasis">z</em>,</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p43"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0479" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>10</mn><mi>y</mi><mo>+</mo><mn>31</mn><mi>z</mi><mo>=</mo><mn>36</mn></mtd></mtr><mtr><mtd><mn>10</mn><mi>y</mi><mo>+</mo><mn>16</mn><mi>z</mi><mo>=</mo><mn>21</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p44">Multiply the first equation by −1 as a means to eliminate the variable <em class="emphasis">y</em>.</p>
<div class="informalfigure large">
<img src="section_06/95d63ad5a9baddc000a1cc8f24c2cb5f.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s02_p46">Now back substitute to find <em class="emphasis">y</em>.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p47"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0480" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>10</mn><mi>y</mi><mo>+</mo><mn>31</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>36</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>y</mi><mo>+</mo><mn>31</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>36</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>y</mi><mo>+</mo><mn>31</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>36</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>5</mn><mrow><mn>10</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p48">Choose any one of the original equations to find <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p49"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0481" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>7</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>6</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>7</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>+</mo><mn>7</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s02_p50">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0482" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch03-s04_s02_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch03_s04_s02_p51"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0483" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mi> </mi><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>17</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s04_s02_p52">Answer: (3, −1, 2)</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/CjSv8D3g2Ic" condition="http://img.youtube.com/vi/CjSv8D3g2Ic/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/CjSv8D3g2Ic" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch03_s04_s03" version="5.0" lang="en">
<h2 class="title editable block">Dependent and Inconsistent Systems</h2>
<p class="para editable block" id="fwk-redden-ch03_s04_s03_p01">Just as with linear systems with two variables, not all linear systems with three variables have a single solution. Sometimes there are no simultaneous solutions.</p>
<div class="callout block" id="fwk-redden-ch03-s04_s03_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch03_s04_s03_p02">Solve the system: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0484" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>4</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>21</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>18</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>9</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>9</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mo>−</mo><mn>8</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p03">In this case we choose to eliminate the variable <em class="emphasis">y</em>.</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p04"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0485" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd><mtable columnspacing="0.1em"><mtr><mtd><mrow><mstyle color="#007fbf"><mi>(</mi><mi>1</mi><mi>)</mi></mstyle></mrow></mtd></mtr><mtr><mtd><mrow><mstyle color="#007fbf"><mi>(</mi><mi>3</mi><mi>)</mi></mstyle></mrow></mtd></mtr></mtable></mtd><mtd><mrow><munder accentunder="true"><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>4</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>5</mn><mtext> </mtext></mtd></mtr><mtr><mtd><mo>−</mo><mn>9</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>9</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>8</mn></mtd></mtr></mtable></mrow><mo stretchy="true">–</mo></munder></mrow></mtd><mtd></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd><mtd><mstyle color="#007f3f"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s03_p05">Next use equations 2 and 3 to eliminate <em class="emphasis">y</em> again.</p>
<div class="informalfigure large">
<img src="section_06/5bb34b567d214f6483e9deac2397a57c.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s03_p07">This leaves a system of two equations with two variables <em class="emphasis">x</em> and <em class="emphasis">z</em>,</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p08"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0486" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mo>−</mo><mn>5</mn><mi>x</mi><mo>−</mo><mtext> </mtext><mn>6</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>15</mn><mi>x</mi><mo>−</mo><mn>18</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>25</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s03_p09">Multiply the first equation by −3 and eliminate the variable <em class="emphasis">z</em>.</p>
<div class="informalfigure large">
<img src="section_06/740c553bef9233cde375b618839805b1.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s03_p11">Adding the resulting equations together leads to a false statement, which indicates that the system is inconsistent. There is no simultaneous solution.</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p12">Answer: Ø</p>
</div>
<p class="para editable block" id="fwk-redden-ch03_s04_s03_p13">A system with no solutions is an inconsistent system. Given three planes, no simultaneous solution can occur in a number of ways.</p>
<div class="informalfigure large block">
<img src="section_06/d550ca0de7c5e923812a078819bc153d.png">
</div>
<p class="para editable block" id="fwk-redden-ch03_s04_s03_p15">Just as with linear systems with two variables, some linear systems with three variables have infinitely many solutions. Such systems are called dependent systems.</p>
<div class="callout block" id="fwk-redden-ch03-s04_s03_n02">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch03_s04_s03_p16">Solve the system: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0487" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd><mspace width="1em"></mspace><mn>7</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi><mo>=</mo><mo>−</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>5</mn><mi>x</mi><mo>+</mo><mn>12</mn><mi>y</mi><mo>−</mo><mn>5</mn><mi>z</mi><mo>=</mo><mo>−</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p17">Eliminate <em class="emphasis">z</em> by adding the first and second equations together.</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p18"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0488" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd><mtable columnspacing="0.1em"><mtr><mtd><mrow><mstyle color="#007fbf"><mi>(</mi><mi>1</mi><mi>)</mi></mstyle></mrow><mtext> </mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>(</mi><mi>2</mi><mi>)</mi></mstyle></mrow></mtd></mtr></mtable></mtd><mtd><mrow><munder accentunder="true"><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>7</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi><mo>=</mo><mo>−</mo><mn>5</mn></mtd></mtr></mtable></mrow><mo stretchy="true">–</mo></munder></mrow></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd columnalign="right"><mtext> </mtext><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>20</mn></mtd><mtd><mtext> </mtext><mstyle color="#007f3f"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s03_p19">Next use equations 1 and 3 to eliminate <em class="emphasis">z</em> again.</p>
<div class="informalfigure large">
<img src="section_06/670a2b19d8ed509edab8ce48b9d5376b.png">
</div>
<p class="para" id="fwk-redden-ch03_s04_s03_p21">This leaves a system of two equations with two variables <em class="emphasis">x</em> and <em class="emphasis">y</em>,</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p22"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0489" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>20</mn></mtd></mtr><mtr><mtd><mn>40</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>80</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s03_p23">Line up the variable <em class="emphasis">y</em> to eliminate by dividing the first equation by 2 and the second equation by −8.</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p24"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0490" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>20</mn></mtd></mtr><mtr><mtd><mn>40</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>80</mn></mtd></mtr></mtable></mrow></mtd><mtd><mtable columnspacing="0.1em"><mtr><mtd><mrow><mover><mo>⇒</mo><mrow><mo>÷</mo><mn>2</mn></mrow></mover></mrow></mtd></mtr><mtr><mtd><mrow><munder><mo>⇒</mo><mrow><mo>÷</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></munder></mrow><mtext> </mtext></mtd></mtr></mtable></mtd><mtd><mrow><munder accentunder="true"><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mspace width="0.8em"></mspace><mn>5</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>10</mn></mtd></mtr></mtable></mrow><mo stretchy="true">–</mo></munder></mrow></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd></mtd><mtd columnalign="right"><mn>0</mn><mo>=</mo><mn>0</mn><mtext> </mtext></mtd><mtd><mtext> </mtext><mstyle color="#007fbf"><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s03_p25">A true statement indicates that the system is dependent. To express the infinite number of solutions <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0491" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></math></span> in terms of one variable, we solve for <em class="emphasis">y</em> and <em class="emphasis">z</em> both in terms of <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p26"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0492" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>20</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn><mi>x</mi><mo>−</mo><mn>20</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mo>−</mo><mn>2</mn><mi>y</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>10</mn><mi>x</mi><mo>−</mo><mn>20</mn></mrow><mrow><mo>−</mo><mn>2</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>+</mo><mn>10</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s03_p27">Once we have <em class="emphasis">y</em> in terms of <em class="emphasis">x</em>, we can solve for <em class="emphasis">z</em> in terms of <em class="emphasis">x</em> by back substituting into one of the original equations.</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p28"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0493" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>5</mn><mi>x</mi><mo>+</mo><mn>10</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>−</mo><mn>20</mn><mi>x</mi><mo>−</mo><mn>40</mn><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>13</mn><mi>x</mi><mo>−</mo><mn>40</mn><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>13</mn><mi>x</mi><mo>+</mo><mn>25</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s03_p29">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0494" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>10</mn><mo>,</mo><mn>13</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow><mo>)</mo></mrow></mrow></math></span>.</p>
</div>
<p class="para editable block" id="fwk-redden-ch03_s04_s03_p30">A consistent system with infinitely many solutions is a dependent system. Given three planes, infinitely many simultaneous solutions can occur in a number of ways.</p>
<div class="informalfigure large block">
<img src="section_06/fba9f2933c40eda5a889e37a0e9c0abf.png">
</div>
<div class="callout block" id="fwk-redden-ch03-s04_s03_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch03_s04_s03_p32"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0495" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>21</mn><mi>x</mi><mo>−</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>8</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s04_s03_p33">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0496" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mfrac><mn>7</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>4</mn><mo>,</mo><mfrac><mrow><mn>14</mn></mrow><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/WgGAtNiJMlI" condition="http://img.youtube.com/vi/WgGAtNiJMlI/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/WgGAtNiJMlI" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch03_s04_s04" version="5.0" lang="en">
<h2 class="title editable block">Applications Involving Three Unknowns</h2>
<p class="para editable block" id="fwk-redden-ch03_s04_s04_p01">Many real-world applications involve more than two unknowns. When an application requires three variables, we look for relationships between the variables that allow us to write three equations.</p>
<div class="callout block" id="fwk-redden-ch03-s04_s04_n01">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch03_s04_s04_p02">A community theater sold 63 tickets to the afternoon performance for a total of $444. An adult ticket cost $8, a child ticket cost $4, and a senior ticket cost $6. If twice as many tickets were sold to adults as to children and seniors combined, how many of each ticket were sold?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p03">Begin by identifying three variables.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p04">Let <em class="emphasis">x</em> represent the number of adult tickets sold.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p05">Let <em class="emphasis">y</em> represent the number of child tickets sold.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p06">Let <em class="emphasis">z</em> represent the number of senior tickets sold.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p07">The first equation comes from the statement that 63 tickets were sold.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p08"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0497" display="block"><mrow><mstyle color="#007fbf"><mi>(</mi><mi>1</mi><mi>)</mi></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>63</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s04_p09">The second equation comes from total ticket sales.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p10"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0498" display="block"><mrow><mstyle color="#007fbf"><mi>(</mi><mi>2</mi><mi>)</mi></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>8</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>z</mi><mo>=</mo><mn>444</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s04_p11">The third equation comes from the statement that twice as many adult tickets were sold as child and senior tickets combined.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p12"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0499" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mi>z</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mi>(</mi><mi>3</mi><mi>)</mi></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s04_p13">Therefore, the problem is modeled by the following linear system.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p14"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0500" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>63</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>444</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s04_s04_p15">Solving this system is left as an exercise. The solution is <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0501" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>42</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s04_s04_p16">Answer: The theater sold 42 adult tickets, 9 child tickets, and 12 senior tickets.</p>
</div>
<div class="key_takeaways editable block" id="fwk-redden-ch03_s04_s04_n02">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch03_s04_s04_l01" mark="bullet">
<li>A simultaneous solution to a linear system with three equations and three variables is an ordered triple (<em class="emphasis">x</em>, <em class="emphasis">y</em>, <em class="emphasis">z</em>) that satisfies all of the equations. If it does not solve each equation, then it is not a solution.</li>
<li>We can solve systems of three linear equations with three unknowns by elimination. Choose any two of the equations and eliminate a variable. Next choose any other two equations and eliminate the same variable. This will result in a system of two equations with two variables that can be solved by any method learned previously.</li>
<li>If the process of solving a system leads to a false statement, then the system is inconsistent and has no solution.</li>
<li>If the process of solving a system leads to a true statement, then the system is dependent and has infinitely many solutions.</li>
<li>To solve applications that require three variables, look for relationships between the variables that allow you to write three linear equations.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch03_s04_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s04_qs01_qd01">
<h3 class="title">Part A: Linear Systems with Three Variables</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s04_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch03_s04_qs01_p01"><strong class="emphasis bold">Determine whether or not the given ordered triple is a solution to the given system.</strong></p>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0502" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0503" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0504" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>8</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0505" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>9</mn><mi>y</mi><mo>+</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0506" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>9</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0507" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>8</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>19</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>9</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>28</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0508" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0509" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0510" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0511" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>12</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mn>9</mn><mi>y</mi><mo>+</mo><mn>8</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0512" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0513" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>−</mo><mn>8</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0514" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0515" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>22</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>13</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0516" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0517" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>y</mi><mo>−</mo><mn>6</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>18</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>9</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0518" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0519" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>a</mi><mo>−</mo><mi>b</mi><mo>+</mo><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>a</mi><mo>−</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0520" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>,</mo><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span>;</p>
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0521" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn><mi>a</mi><mo>+</mo><mn>2</mn><mi>b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>25</mn><mi>a</mi><mo>+</mo><mn>5</mn><mi>b</mi><mo>+</mo><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch03_s04_qs01_qd02">
<h3 class="title">Part B: Solving Linear Systems with Three Variables</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s04_qs01_qd02_qd01" start="11">
<p class="para" id="fwk-redden-ch03_s04_qs01_p22"><strong class="emphasis bold">Solve.</strong></p>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa11">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0522" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa12">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0523" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa13">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0524" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>21</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa14">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0525" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa15">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0526" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>5</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>12</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa16">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0527" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>17</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa17">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0528" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa18">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0529" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>11</mn><mi>x</mi><mo>+</mo><mn>18</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa19">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0530" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa20">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0531" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa21">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0532" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa22">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0534" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>8</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa23">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0536" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>11</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa24">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0538" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa25">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0540" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa26">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0542" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>12</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>−</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa27">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0544" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>a</mi><mo>−</mo><mi>b</mi><mo>+</mo><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>a</mi><mo>−</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa28">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0545" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn><mi>a</mi><mo>+</mo><mn>2</mn><mi>b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>25</mn><mi>a</mi><mo>+</mo><mn>5</mn><mi>b</mi><mo>+</mo><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa29">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0546" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>22</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>x</mi><mo>+</mo><mn>8</mn><mi>y</mi><mo>−</mo><mn>5</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>20</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa30">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0547" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>−</mo><mn>7</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa31">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0548" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>9</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>−</mo><mn>6</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>11</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa32">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0549" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa33">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0550" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa34">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0551" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>28</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>y</mi><mo>−</mo><mn>7</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>27</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa35">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0552" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa36">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0553" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>6</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa37">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0555" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>30</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa38">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0556" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>4</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa39">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0558" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>12</mn><mi>x</mi><mo>+</mo><mn>16</mn><mi>y</mi><mo>+</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>+</mo><mn>13</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa40">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0560" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>8</mn><mi>x</mi><mo>+</mo><mn>12</mn><mi>y</mi><mo>−</mo><mn>8</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa41">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0562" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>17</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>−</mo><mn>2</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>−</mo><mn>4</mn><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>13</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa42">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0564" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>7</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>−</mo><mi>y</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mi>z</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>11</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa43">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0566" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><mi>a</mi><mo>−</mo><mn>2</mn><mi>b</mi><mo>+</mo><mn>3</mn><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>a</mi><mo>+</mo><mn>3</mn><mi>b</mi><mo>−</mo><mn>5</mn><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>a</mi><mo>−</mo><mn>6</mn><mi>b</mi><mo>+</mo><mn>5</mn><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>13</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa44">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0567" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>6</mn><mi>a</mi><mo>−</mo><mn>2</mn><mi>b</mi><mo>+</mo><mn>5</mn><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>a</mi><mo>+</mo><mn>3</mn><mi>b</mi><mo>−</mo><mn>3</mn><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>a</mi><mo>+</mo><mn>5</mn><mi>b</mi><mo>+</mo><mn>6</mn><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>24</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch03_s04_qs01_qd03">
<h3 class="title">Part C: Applications</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s04_qs01_qd03_qd01" start="45">
<p class="para" id="fwk-redden-ch03_s04_qs01_p91"><strong class="emphasis bold">Set up a system of equations and use it to solve the following.</strong></p>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p92">The sum of three integers is 38. Two less than 4 times the smaller integer is equal to the sum of the others. The sum of the smaller and larger integer is equal to 2 more than twice that of the other. Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p94">The sum of three integers is 40. Three times the smaller integer is equal to the sum of the others. Twice the larger is equal to 8 more than the sum of the others. Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p96">The sum of the angles <em class="emphasis">A</em>, <em class="emphasis">B</em>, and <em class="emphasis">C</em> of a triangle is 180°. The larger angle <em class="emphasis">C</em> is equal to twice the sum of the other two. Four times the smallest angle <em class="emphasis">A</em> is equal to the difference of angle <em class="emphasis">C</em> and <em class="emphasis">B</em>. Find the angles.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p98">The sum of the angles <em class="emphasis">A</em>, <em class="emphasis">B</em>, and <em class="emphasis">C</em> of a triangle is 180°. Angle <em class="emphasis">C</em> is equal to the sum of the other two angles. Five times angle <em class="emphasis">A</em> is equal to the sum of angle <em class="emphasis">C</em> and <em class="emphasis">B</em>. Find the angles.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p100">A total of $12,000 was invested in three interest earning accounts. The interest rates were 2%, 4%, and 5%. If the total simple interest for one year was $400 and the amount invested at 2% was equal to the sum of the amounts in the other two accounts, then how much was invested in each account?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p102">Joe invested his $6,000 bonus in three accounts earning <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0568" display="inline"><mrow><mn>4</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>% interest. He invested twice as much in the account earning <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0569" display="inline"><mrow><mn>4</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>% as he did in the other two accounts combined. If the total simple interest for the year was $234, how much did Joe invest in each account?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p104">A jar contains nickels, dimes, and quarters. There are 105 coins with a total value of $8.40. If there are 3 more than twice as many dimes as quarters, find how many of each coin are in the jar.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p106">A billfold holds one-dollar, five-dollar, and ten-dollar bills and has a value of $210. There are 50 bills total where the number of one-dollar bills is one less than twice the number of five-dollar bills. How many of each bill are there?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p108">A nurse wishes to prepare a 15-ounce topical antiseptic solution containing 3% hydrogen peroxide. To obtain this mixture, purified water is to be added to the existing 1.5% and 10% hydrogen peroxide products. If only 3 ounces of the 10% hydrogen peroxide solution is available, how much of the 1.5% hydrogen peroxide solution and water is needed?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p110">A chemist needs to produce a 32-ounce solution consisting of <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0572" display="inline"><mrow><mn>8</mn><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span>% acid. He has three concentrates with 5%, 10%, and 40% acid. If he is to use twice as much of the 5% acid solution as the 10% solution, then how many ounces of the 40% solution will he need?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p112">A community theater sold 128 tickets to the evening performance for a total of $1,132. An adult ticket cost $10, a child ticket cost $5, and a senior ticket cost $6. If three times as many tickets were sold to adults as to children and seniors combined, how many of each ticket were sold?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p114">James sold 82 items at the swap meet for a total of $504. He sold packages of socks for $6, printed t-shirts for $12, and hats for $5. If he sold 5 times as many hats as he did t-shirts, how many of each item did he sell?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p116">A parabola passes through three points <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0573" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0574" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0575" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>. Use these points and <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0576" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> to construct a system of three linear equations in terms of <em class="emphasis">a</em>, <em class="emphasis">b</em>, and <em class="emphasis">c</em> and then solve the system.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p118">A parabola passes through three points <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0580" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>11</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0581" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0582" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>. Use these points and <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0583" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> to construct a system of three linear equations in terms of <em class="emphasis">a</em>, <em class="emphasis">b</em>, and <em class="emphasis">c</em> and solve it.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch03_s04_qs01_qd04">
<h3 class="title">Part D: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s04_qs01_qd04_qd01" start="59">
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p120">On a note card, write down the steps for solving a system of three linear equations with three variables using elimination. Use your notes to explain to a friend how to solve one of the exercises in this section.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch03_s04_qs01_p121">Research and discuss curve fitting. Why is curve fitting an important topic?</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch03_s04_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p03_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p07_ans">Yes</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p11_ans">Yes</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p15_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p19_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="11">
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p24_ans">(2, −1, −3)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p28_ans">(4, 1, −3)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p32_ans">(1, −1, 3)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p36_ans">Ø</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p40_ans">(5, −10, −6)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa21_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0533" display="block"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa23_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0537" display="block"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mtext> </mtext><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa24_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa25_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0541" display="block"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mn>3</mn><mo>,</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa26_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa27_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p56_ans">(1, −2, 6)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa28_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa29_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p60_ans">(−1, 2, −2)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa30_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa31_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p64_ans">(1,−3, 5)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa32_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa33_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p68_ans">(1, 1, 0)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa34_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa35_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p72_ans">(0, 1, −2)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa36_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa37_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p76_ans">(5, 0, 6)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa38_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa39_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p80_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0559" display="inline"><mo>Ø</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa40_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa41_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p84_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0563" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa42_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa43_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p88_ans">(1, 2, 3)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa44_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="45">
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa45_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p93_ans">8, 12, 18</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa46_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa47_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p97_ans"><em class="emphasis">A</em> = 20°, <em class="emphasis">B</em> = 40°, and <em class="emphasis">C</em> = 120°</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa48_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa49_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p101_ans">The amount invested at 2% was $6,000, the amount invested at 4% was $2,000, and the amount invested at 5% was $4,000.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa50_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa51_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p105_ans">72 nickels, 23 dimes, and 10 quarters</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa52_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa53_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p109_ans">10 ounces of the 1.5% hydrogen peroxide solution and 2 ounces of water</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa54_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa55_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p113_ans">96 adult tickets, 20 child tickets, and 12 senior tickets were sold.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa56_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa57_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s04_qs01_p117_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0577" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0578" display="inline"><mrow><mi>b</mi><mo>=</mo><mo>−</mo><mn>4</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0579" display="inline"><mrow><mi>c</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa58_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="59">
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa59_ans">
<div class="answer">
<p class="para">Answer may vary</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s04_qs01_qa60_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
</div>
</div>
</div>
</div>
<div id=navbar-bottom class="navbar">
<div class="navbar-part left">
<a href="s06-03-applications-of-linear-systems.html"><img src="shared/images/batch-left.png"></a> <a href="s06-03-applications-of-linear-systems.html">Previous Section</a>
</div>
<div class="navbar-part middle">
<a href="index.html"><img src="shared/images/batch-up.png"></a> <a href="index.html">Table of Contents</a>
</div>
<div class="navbar-part right">
<a href="s06-05-matrices-and-gaussian-eliminat.html">Next Section</a> <a href="s06-05-matrices-and-gaussian-eliminat.html"><img src="shared/images/batch-right.png"></a>
</div>
</div>
</div>
<script type="text/javascript" src="shared/book.js"></script>
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=MML_HTMLorMML"></script>
</body>
</html>