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<div class="section" id="fwk-redden-ch01_s07" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">1.7</span> Solving Linear Equations</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch01_s07_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch01_s07_o01" numeration="arabic">
<li>Use the properties of equality to solve basic linear equations.</li>
<li>Identify and solve conditional linear equations, identities, and contradictions.</li>
<li>Clear fractions from equations.</li>
<li>Set up and solve linear applications.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch01_s07_s01" version="5.0" lang="en">
<h2 class="title editable block">Solving Basic Linear Equations</h2>
<p class="para block" id="fwk-redden-ch01_s07_s01_p01">An <span class="margin_term"><a class="glossterm">equation</a><span class="glossdef">Statement indicating that two algebraic expressions are equal.</span></span> is a statement indicating that two algebraic expressions are equal. A <span class="margin_term"><a class="glossterm">linear equation with one variable</a><span class="glossdef">An equation that can be written in the standard form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1738" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>0</mi></mrow></math></span>, where <em class="emphasis">a</em> and <em class="emphasis">b</em> are real numbers and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1739" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span>, <em class="emphasis">x</em>, is an equation that can be written in the standard form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1740" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>0</mi></mrow></math></span> where <em class="emphasis">a</em> and <em class="emphasis">b</em> are real numbers and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1741" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span> For example,</p>
<p class="para block" id="fwk-redden-ch01_s07_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1742" display="block"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>12</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s07_s01_p03">A <span class="margin_term"><a class="glossterm">solution</a><span class="glossdef">Any value that can replace the variable in an equation to produce a true statement.</span></span> to a linear equation is any value that can replace the variable to produce a true statement. The variable in the linear equation <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1743" display="inline"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>12</mn><mo>=</mo><mn>0</mn></mrow></math></span> is <em class="emphasis">x</em> and the solution is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1744" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span> To verify this, substitute the value 4 in for <em class="emphasis">x</em> and check that you obtain a true statement.</p>
<p class="para block" id="fwk-redden-ch01_s07_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1745" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>12</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mo stretchy="false">(</mo><mstyle color="#007fbf"><mn>4</mn></mstyle><mo stretchy="false">)</mo><mo>−</mo><mn>12</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>12</mn><mo>−</mo><mn>12</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>✓</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s07_s01_p05">Alternatively, when an equation is equal to a constant, we may verify a solution by substituting the value in for the variable and showing that the result is equal to that constant. In this sense, we say that solutions “satisfy the equation.”</p>
<div class="callout block" id="fwk-redden-ch01_s07_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch01_s07_s01_p06">Is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1746" display="inline"><mrow><mi>a</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> a solution to <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1747" display="inline"><mrow><mo>−</mo><mn>10</mn><mi>a</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>25</mn></mrow></math></span>?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s01_p07">Recall that when evaluating expressions, it is a good practice to first replace all variables with parentheses, and then substitute the appropriate values. By making use of parentheses, we avoid some common errors when working the order of operations.</p>
<p class="para" id="fwk-redden-ch01_s07_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1748" display="block"><mrow><mo>−</mo><mn>10</mn><mi>a</mi><mo>+</mo><mn>5</mn><mo>=</mo><mo>−</mo><mn>10</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>5</mn><mo>=</mo><mn>10</mn><mo>≠</mo><mn>25</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s01_p09">Answer: No, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1749" display="inline"><mrow><mi>a</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> does not satisfy the equation.</p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s01_p10">Developing techniques for solving various algebraic equations is one of our main goals in algebra. This section reviews the basic techniques used for solving linear equations with one variable. We begin by defining <span class="margin_term"><a class="glossterm">equivalent equations</a><span class="glossdef">Equations with the same solution set.</span></span> as equations with the same solution set.</p>
<p class="para block" id="fwk-redden-ch01_s07_s01_p11"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1750" display="block"><mrow><mrow><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn><mo>=</mo><mn>16</mn><mtext> </mtext></mrow></mtd></mtr><mtr><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>3</mn><mi>x</mi><mo>=</mo><mn>21</mn></mrow></mtd></mtr><mtr><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi><mo>=</mo><mn>7</mn></mrow></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mi> </mi><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>E</mi><mi>q</mi><mi>u</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi> </mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></mstyle></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s07_s01_p12">Here we can see that the three linear equations are equivalent because they share the same solution set, namely, {7}. To obtain equivalent equations, use the following <span class="margin_term"><a class="glossterm">properties of equality</a><span class="glossdef">Properties that allow us to obtain equivalent equations by adding, subtracting, multiplying, and dividing both sides of an equation by nonzero real numbers.</span></span>. Given algebraic expressions <em class="emphasis">A</em> and <em class="emphasis">B</em>, where <em class="emphasis">c</em> is a nonzero number:</p>
<p class="para block" id="fwk-redden-ch01_s07_s01_p13">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Addition property of equality:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1751" display="inline"><mrow><mtext>If</mtext><mi> </mi><mi>A</mi><mo>=</mo><mi>B</mi><mo>,</mo><mi> </mi><mi> </mi><mtext>then</mtext><mi> </mi><mi> </mi><mi>A</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo><mtext> </mtext><mi>c</mi></mstyle><mo>=</mo><mi>B</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo><mtext> </mtext><mi>c</mi></mstyle></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Subtraction property of equality:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1752" display="inline"><mrow><mtext>If</mtext><mi> </mi><mi>A</mi><mo>=</mo><mi>B</mi><mo>,</mo><mi> </mi><mi> </mi><mtext>then</mtext><mi> </mi><mi> </mi><mi>A</mi><mstyle color="#007fbf"><mo>−</mo><mi>c</mi></mstyle><mo>=</mo><mi>B</mi><mstyle color="#007fbf"><mo>−</mo><mi>c</mi></mstyle></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Multiplication property of equality:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1753" display="inline"><mrow><mtext>If</mtext><mi> </mi><mi>A</mi><mo>=</mo><mi>B</mi><mo>,</mo><mi> </mi><mi> </mi><mtext>then</mtext><mi> </mi><mi> </mi><mstyle color="#007fbf"><mi>c</mi></mstyle><mi>A</mi><mo>=</mo><mstyle color="#007fbf"><mi>c</mi></mstyle><mi>B</mi><mtext> </mtext></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Division property of equality:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1754" display="inline"><mrow><mtext>If</mtext><mi> </mi><mi>A</mi><mo>=</mo><mi>B</mi><mo>,</mo><mi> </mi><mi> </mi><mtext>then</mtext><mi> </mi><mtext> </mtext><mfrac><mi>A</mi><mstyle color="#007fbf"><mi>c</mi></mstyle></mfrac><mo>=</mo><mfrac><mi>B</mi><mstyle color="#007fbf"><mi>c</mi></mstyle></mfrac></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s01_p14"><strong class="emphasis bold">Note:</strong> Multiplying or dividing both sides of an equation by 0 is carefully avoided. Dividing by 0 is undefined and multiplying both sides by 0 results in the equation 0 = 0.</p>
<p class="para block" id="fwk-redden-ch01_s07_s01_p15">We solve algebraic equations by isolating the variable with a coefficient of 1. If given a linear equation of the form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1755" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>c</mi></mrow></math></span>, then we can solve it in two steps. First, use the appropriate equality property of addition or subtraction to isolate the variable term. Next, isolate the variable using the equality property of multiplication or division. Checking the solution in the following examples is left to the reader.</p>
<div class="callout block" id="fwk-redden-ch01_s07_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch01_s07_s01_p16">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1756" display="inline"><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>19</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s01_p17"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1757" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>−</mo><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>19</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>−</mo><mn>2</mn><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>2</mn></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>19</mn><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>2</mn></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>A</mi><mi>d</mi><mi>d</mi><mtext> </mtext><mn>2</mn><mtext> </mtext><mi>t</mi><mi>o</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo><mtext> </mtext></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>21</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>7</mn><mi>x</mi></mrow><mstyle color="#007fbf"><mn>7</mn></mstyle></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>21</mn></mrow><mstyle color="#007fbf"><mn>7</mn></mstyle></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mn>7</mn><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s01_p18">Answer: The solution is 3.</p>
</div>
<div class="callout block" id="fwk-redden-ch01_s07_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch01_s07_s01_p19">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1758" display="inline"><mrow><mn>56</mn><mo>=</mo><mn>8</mn><mo>+</mo><mn>12</mn><mi>y</mi></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s01_p20">When no sign precedes the term, it is understood to be positive. In other words, think of this as <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1759" display="inline"><mrow><mn>56</mn><mo>=</mo><mo>+</mo><mn>8</mn><mo>+</mo><mn>12</mn><mi>y</mi></mrow><mo>.</mo></math></span> Therefore, we begin by subtracting 8 on both sides of the equal sign.</p>
<p class="para" id="fwk-redden-ch01_s07_s01_p21"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1760" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>56</mn><mstyle color="#007fbf"><mo>−</mo><mn>8</mn></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn><mo>+</mo><mn>12</mn><mi>y</mi><mstyle color="#007fbf"><mo>−</mo><mn>8</mn></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>48</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>12</mn><mi>y</mi></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>48</mn></mrow><mrow><mstyle color="#007fbf"><mn>12</mn></mstyle></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>12</mn><mi>y</mi></mrow><mstyle color="#007fbf"><mrow><mn>12</mn></mrow></mstyle></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>y</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s01_p22">It does not matter on which side we choose to isolate the variable because the <span class="margin_term"><a class="glossterm">symmetric property</a><span class="glossdef">Allows you to solve for the variable on either side of the equal sign, because <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1761" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>5</mn></mrow></math></span> is equivalent to <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1762" display="inline"><mrow><mn>5</mn><mo>=</mo><mi>x</mi></mrow><mo>.</mo></math></span></span></span> states that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1763" display="inline"><mrow><mn>4</mn><mo>=</mo><mi>y</mi></mrow></math></span> is equivalent to <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1764" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s01_p23">Answer: The solution is 4.</p>
</div>
<div class="callout block" id="fwk-redden-ch01_s07_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch01_s07_s01_p24">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1765" display="inline"><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mo>−</mo><mn>8</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s01_p25">Isolate the variable term using the addition property of equality, and then multiply both sides of the equation by the reciprocal of the coefficient <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1766" display="inline"><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s01_p26"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1767" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>S</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mn>2</mn><mtext> </mtext><mi>o</mi><mi>n</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mfrac><mn>3</mn><mn>5</mn></mfrac></mstyle><mo>⋅</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mfrac><mn>3</mn><mrow><menclose notation="updiagonalstrike"><mn>5</mn></menclose></mrow></mfrac></mstyle><mo>⋅</mo><mover><mrow><mrow><mo>(</mo><mrow><menclose notation="updiagonalstrike"><mrow><mo>−</mo><mn>10</mn></mrow></menclose></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></mover><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>M</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext></mrow><mfrac><mn>3</mn><mn>5</mn></mfrac><mtext>.</mtext></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>1</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mo>⋅</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>6</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s01_p27">Answer: The solution is −6.</p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s01_p28">In summary, to retain equivalent equations, we must perform the same operation on both sides of the equation.</p>
<div class="callout block" id="fwk-redden-ch01_s07_s01_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch01_s07_s01_p29"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1768" display="inline"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s01_p30">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1769" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span></p>
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</div>
</div>
</div>
<div class="section" id="fwk-redden-ch01_s07_s02" version="5.0" lang="en">
<h2 class="title editable block">General Guidelines for Solving Linear Equations</h2>
<p class="para editable block" id="fwk-redden-ch01_s07_s02_p01">Typically linear equations are not given in standard form, and so solving them requires additional steps. When solving linear equations, the goal is to determine what value, if any, will produce a true statement when substituted in the original equation. Do this by isolating the variable using the following steps:</p>
<ul class="itemizedlist editable block" id="fwk-redden-ch01_s07_s02_l01" mark="none">
<li>
<strong class="emphasis bold">Step 1:</strong> Simplify both sides of the equation using the order of operations and combine all like terms on the same side of the equal sign.</li>
<li>
<strong class="emphasis bold">Step 2:</strong> Use the appropriate properties of equality to combine like terms on opposite sides of the equal sign. The goal is to obtain the variable term on one side of the equation and the constant term on the other.</li>
<li>
<strong class="emphasis bold">Step 3:</strong> Divide or multiply as needed to isolate the variable.</li>
<li>
<strong class="emphasis bold">Step 4:</strong> Check to see if the answer solves the original equation.</li>
</ul>
<p class="para editable block" id="fwk-redden-ch01_s07_s02_p02">We will often encounter linear equations where the expressions on each side of the equal sign can be simplified. If this is the case, then it is best to simplify each side first before solving. Normally this involves combining same-side like terms.</p>
<p class="para editable block" id="fwk-redden-ch01_s07_s02_p03"><strong class="emphasis bold">Note:</strong> At this point in our study of algebra the use of the properties of equality should seem routine. Therefore, displaying these steps in this text, usually in blue, becomes optional.</p>
<div class="callout block" id="fwk-redden-ch01_s07_s02_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch01_s07_s02_p04">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1770" display="inline"><mrow><mo>−</mo><mn>4</mn><mi>a</mi><mo>+</mo><mn>2</mn><mo>−</mo><mi>a</mi><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p05">First combine the like terms on the left side of the equal sign.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1771" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>4</mn><mi>a</mi><mo>+</mo><mn>2</mn><mo>−</mo><mi>a</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>C</mi><mi>o</mi><mi>m</mi><mi>b</mi><mi>i</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>s</mi><mi>a</mi><mi>m</mi><mi>e</mi><mtext>-</mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mtext> </mtext><mi>l</mi><mi>i</mi><mi>k</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>a</mi><mo>+</mo><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>S</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mn>2</mn><mtext> </mtext><mi>o</mi><mi>n</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>a</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mo>−</mo><mn>5.</mn></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>a</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p07">Always use the original equation to check to see if the solution is correct.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1772" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>4</mn><mi>a</mi><mo>+</mo><mn>2</mn><mo>−</mo><mi>a</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mfrac><mn>1</mn><mn>5</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mo>−</mo><mstyle color="#007f3f"><mfrac><mn>1</mn><mn>5</mn></mfrac></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mo>+</mo><mfrac><mn>2</mn><mn>1</mn></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mfrac><mn>5</mn><mn>5</mn></mfrac></mstyle><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>4</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>1</mn></mrow><mn>5</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>5</mn><mn>5</mn></mfrac><mo>=</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p09">Answer: The solution is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1773" display="inline"><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch01_s07_s02_p10">Given a linear equation in the form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1774" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></mrow></math></span>, we begin the solving process by combining like terms on opposite sides of the equal sign. To do this, use the addition or subtraction property of equality to place like terms on the same side so that they can be combined. In the examples that remain, the check is left to the reader.</p>
<div class="callout block" id="fwk-redden-ch01_s07_s02_n02">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch01_s07_s02_p11">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1775" display="inline"><mrow><mo>−</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>11</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p12">Subtract <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1776" display="inline"><mrow><mn>5</mn><mi>y</mi></mrow></math></span> on both sides so that we can combine the terms involving <em class="emphasis">y</em> on the left side.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p13"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1777" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>3</mn><mstyle color="#007fbf"><mo>−</mo><mn>5</mn><mi>y</mi></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mi>y</mi><mo>+</mo><mn>11</mn><mstyle color="#007fbf"><mo>−</mo><mn>5</mn><mi>y</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn><mi>y</mi><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>11</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p14">From here, solve using the techniques developed previously.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1778" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn><mi>y</mi><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>11</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>A</mi><mi>d</mi><mi>d</mi><mtext> </mtext><mn>3</mn><mtext> </mtext><mi>t</mi><mi>o</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn><mi>y</mi><mi> </mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi> </mi><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>14</mn></mrow><mrow><mo>−</mo><mn>7</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mo>−</mo><mn>7.</mn></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi><mi> </mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi> </mi><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p16">Answer: The solution is −2.</p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s02_p17">Solving will often require the application of the distributive property.</p>
<div class="callout block" id="fwk-redden-ch01_s07_s02_n03">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch01_s07_s02_p18">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1779" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mn>7</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p19">Simplify the linear expressions on either side of the equal sign first.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p20"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1780" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow><mstyle color="#007fbf"><mtext> </mtext><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>+</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn><mo>−</mo><mn>14</mn><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>C</mi><mi>o</mi><mi>m</mi><mi>b</mi><mi>i</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>s</mi><mi>a</mi><mi>m</mi><mi>e</mi><mtext>-</mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mtext> </mtext><mi>l</mi><mi>i</mi><mi>k</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn><mo>−</mo><mn>14</mn><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>C</mi><mi>o</mi><mi>m</mi><mi>b</mi><mi>i</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>o</mi><mi>p</mi><mi>p</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>e</mi><mtext>-</mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mtext> </mtext><mi>l</mi><mi>i</mi><mi>k</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>3</mn><mn>9</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p21">Answer: The solution is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1781" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s07_s02_n04">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch01_s07_s02_p22">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1782" display="inline"><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mi>a</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>a</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p23">Begin by applying the distributive property.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p24"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1783" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mi>a</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>a</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>15</mn><mo>−</mo><mn>5</mn><mi>a</mi><mo>−</mo><mn>10</mn><mo>+</mo><mn>4</mn><mi>a</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mo>−</mo><mi>a</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mi>a</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-ch01_s07_s02_p25">Here we point out that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1784" display="inline"><mrow><mo>−</mo><mi>a</mi></mrow></math></span> is equivalent to <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1785" display="inline"><mrow><mo>−</mo><mn>1</mn><mi>a</mi></mrow></math></span>; therefore, we choose to divide both sides of the equation by −1.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p26"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1786" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mo>−</mo><mi>a</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mo>−</mo><mn>1</mn><mi>a</mi></mrow><mrow><mstyle color="#007fbf"><mo>−</mo><mn>1</mn></mstyle></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mrow><mstyle color="#007fbf"><mo>−</mo><mn>1</mn></mstyle></mrow></mfrac></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p27">Alternatively, we can multiply both sides of <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1787" display="inline"><mrow><mo>−</mo><mi>a</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span> by negative one and achieve the same result.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p28"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1788" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mo>−</mo><mi>a</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd><mrow><mstyle color="#007fbf"><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow></mstyle><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>a</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mstyle color="#007fbf"><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow></mstyle><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p29">Answer: The solution is 2.</p>
</div>
<div class="callout block" id="fwk-redden-ch01_s07_s02_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch01_s07_s02_p30"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1789" display="inline"><mrow><mn>6</mn><mo>−</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p31">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1790" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span></p>
<div class="mediaobject">
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</div>
</div>
<p class="para block" id="fwk-redden-ch01_s07_s02_p33">There are three different types of equations. Up to this point, we have been solving <span class="margin_term"><a class="glossterm">conditional equations</a><span class="glossdef">Equations that are true for particular values.</span></span>. These are equations that are true for particular values. An <span class="margin_term"><a class="glossterm">identity</a><span class="glossdef">An equation that is true for all possible values.</span></span> is an equation that is true for all possible values of the variable. For example,
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1791" display="block"><mrow><mi>x</mi><mi> </mi><mo>=</mo><mi> </mi><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>I</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow></mstyle></mrow></math></span>
has a solution set consisting of all real numbers, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1792" display="inline"><mi>ℝ</mi><mo>.</mo></math></span> A <span class="margin_term"><a class="glossterm">contradiction</a><span class="glossdef">An equation that is never true and has no solution.</span></span> is an equation that is never true and thus has no solutions. For example,
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1793" display="block"><mrow><mi>x</mi><mi> </mi><mo>+</mo><mi> </mi><mn>1</mn><mi> </mi><mo>=</mo><mi> </mi><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>C</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow></mstyle></mrow></math></span>
has no solution. We use the empty set, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1794" display="inline"><mo>Ø</mo></math></span>, to indicate that there are no solutions.</p>
<p class="para editable block" id="fwk-redden-ch01_s07_s02_p38">If the end result of solving an equation is a true statement, like 0 = 0, then the equation is an identity and any real number is a solution. If solving results in a false statement, like 0 = 1, then the equation is a contradiction and there is no solution.</p>
<div class="callout block" id="fwk-redden-ch01_s07_s02_n05">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch01_s07_s02_p39">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1795" display="inline"><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>6</mn><mo>=</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p40"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1796" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>+</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>+</mo><mn>20</mn><mo>+</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>+</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>+</mo><mn>26</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>+</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>26</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mi>✗</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p41">Solving leads to a false statement; therefore, the equation is a contradiction and there is no solution.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p42">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1797" display="inline"><mo>Ø</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s07_s02_n06">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch01_s07_s02_p43">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1798" display="inline"><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn><mo>=</mo><mn>10</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>y</mi></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p44"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1799" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mo stretchy="false">(</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mi>y</mi><mo>+</mo><mn>15</mn><mo>+</mo><mn>5</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn><mi>y</mi><mo>+</mo><mn>20</mn><mo>−</mo><mi>y</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mi>y</mi><mo>+</mo><mn>20</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn><mi>y</mi><mo>+</mo><mn>20</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn><mi>y</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>✓</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p45">Solving leads to a true statement; therefore, the equation is an identity and any real number is a solution.</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p46">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1800" display="inline"><mi>ℝ</mi></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s02_p47">The coefficients of linear equations may be any real number, even decimals and fractions. When this is the case it is possible to use the multiplication property of equality to clear the fractional coefficients and obtain integer coefficients in a single step. If given fractional coefficients, then multiply both sides of the equation by the least common multiple of the denominators (LCD).</p>
<div class="callout block" id="fwk-redden-ch01_s07_s02_n07">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch01_s07_s02_p48">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1801" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s02_p49">Clear the fractions by multiplying both sides by the least common multiple of the given denominators. In this case, it is the <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1802" display="inline"><mrow><mi>L</mi><mi>C</mi><mi>D</mi><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>15</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p50"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1803" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mn>15</mn></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>15</mn></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>M</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mi>1</mi><mi>5</mi></mrow><mtext>.</mtext></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mn>15</mn></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mstyle color="#007fbf"><mn>15</mn></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>15</mn></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mstyle color="#007fbf"><mn>15</mn></mstyle><mo>⋅</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mtext>.</mtext></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mn>15</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi></mrow><mtext>.</mtext></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>18</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>18</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mo>−</mo><mn>9</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s02_p51">Answer: The solution is −9.</p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s02_p52">It is important to know that this technique only works for equations. <em class="emphasis bolditalic">Do not try to clear fractions when simplifying expressions.</em> As a reminder:</p>
<p class="para block" id="fwk-redden-ch01_s07_s02_p53">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">Expression</p></th>
<th align="center"><p class="para">Equation</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1804" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1805" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s02_p54">We simplify expressions and solve equations. If you multiply an expression by 6, you will change the problem. However, if you multiply both sides of an equation by 6, you obtain an equivalent equation.</p>
<p class="para block" id="fwk-redden-ch01_s07_s02_p55">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">Incorrect</p></th>
<th align="center"><p class="para">Correct</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1806" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mo>≠</mo><mstyle color="#ff0000"><mn>6</mn><mo>⋅</mo></mstyle><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mo>=</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1807" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mn>6</mn><mo>⋅</mo></mstyle><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>6</mn><mo>⋅</mo></mstyle><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>10</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>✓</mi></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="section" id="fwk-redden-ch01_s07_s03" version="5.0" lang="en">
<h2 class="title editable block">Applications Involving Linear Equations</h2>
<p class="para editable block" id="fwk-redden-ch01_s07_s03_p01">Algebra simplifies the process of solving real-world problems. This is done by using letters to represent unknowns, restating problems in the form of equations, and by offering systematic techniques for solving those equations. To solve problems using algebra, first translate the wording of the problem into mathematical statements that describe the relationships between the given information and the unknowns. Usually, this translation to mathematical statements is the difficult step in the process. The key to the translation is to carefully read the problem and identify certain key words and phrases.</p>
<p class="para block" id="fwk-redden-ch01_s07_s03_p02">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="left"><p class="para">Key Words</p></th>
<th align="left"><p class="para">Translation</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Sum</strong>, increased by, more than, plus, added to, total</p></td>
<td align="left"><p class="para"> +</p></td>
</tr>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Difference</strong>, decreased by, subtracted from, less, minus</p></td>
<td align="left"><p class="para"> −</p></td>
</tr>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Product</strong>, multiplied by, of, times, twice</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1808" display="inline"><mo>⋅</mo></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Quotient</strong>, divided by, ratio, per</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1809" display="inline"><mo>÷</mo></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Is</strong>, total, result</p></td>
<td align="left"><p class="para"> =</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s03_p03">When translating sentences into mathematical statements, be sure to read the sentence several times and parse out the key words and phrases. It is important to first identify the variable, “<em class="emphasis bolditalic">let x represent…</em>” and state in words what the unknown quantity is. This step not only makes our work more readable, but also forces us to think about what we are looking for.</p>
<div class="callout block" id="fwk-redden-ch01_s07_s03_n01">
<h3 class="title">Example 12</h3>
<p class="para" id="fwk-redden-ch01_s07_s03_p04">When 6 is subtracted from twice the sum of a number and 8 the result is 5. Find the number.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p05">Let <em class="emphasis">n</em> represent the unknown number.</p>
<div class="informalfigure large">
<img src="section_04/fb5b25de9d7267c4fdc3cf2953eae974.png">
</div>
<p class="para" id="fwk-redden-ch01_s07_s03_p07">To understand why we included the parentheses in the set up, you must study the structure of the following two sentences and their translations:</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p08">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para">“<em class="emphasis bolditalic">twice the sum</em> <em class="emphasis">of a number and 8</em>”</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1810" display="inline"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">“<em class="emphasis">the sum of <strong class="emphasis bold">twice a number</strong> and 8</em>”</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1811" display="inline"><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>8</mn></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch01_s07_s03_p09">The key was to focus on the phrase “<em class="emphasis bolditalic">twice the sum</em>,” this prompted us to group the sum within parentheses and then multiply by 2. After translating the sentence into a mathematical statement we then solve.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p10"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1812" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>n</mi><mo>+</mo><mn>16</mn><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>n</mi><mo>+</mo><mn>10</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>n</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>n</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>5</mn></mrow><mn>2</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p11">Check.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p12"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1813" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>8</mn><mo stretchy="false">)</mo><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mn>11</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>11</mn><mo>−</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>✓</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p13">Answer: The number is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1814" display="inline"><mrow><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s07_s03_p14">General guidelines for setting up and solving word problems follow.</p>
<ul class="itemizedlist editable block" id="fwk-redden-ch01_s07_s03_l01" mark="none">
<li>
<strong class="emphasis bold">Step 1:</strong> Read the problem several times, identify the key words and phrases, and organize the given information.</li>
<li>
<strong class="emphasis bold">Step 2:</strong> Identify the variables by assigning a letter or expression to the unknown quantities.</li>
<li>
<strong class="emphasis bold">Step 3:</strong> Translate and set up an algebraic equation that models the problem.</li>
<li>
<strong class="emphasis bold">Step 4:</strong> Solve the resulting algebraic equation.</li>
<li>
<strong class="emphasis bold">Step 5:</strong> Finally, answer the question in sentence form and make sure it makes sense (check it).</li>
</ul>
<p class="para editable block" id="fwk-redden-ch01_s07_s03_p15">For now, set up all of your equations using only one variable. Avoid two variables by looking for a relationship between the unknowns.</p>
<div class="callout block" id="fwk-redden-ch01_s07_s03_n02">
<h3 class="title">Example 13</h3>
<p class="para" id="fwk-redden-ch01_s07_s03_p16">A rectangle has a perimeter measuring 92 meters. The length is 2 meters less than 3 times the width. Find the dimensions of the rectangle.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p17">The sentence “<em class="emphasis">The length is</em> <em class="emphasis bolditalic">2 meters less</em> <em class="emphasis">than</em> <em class="emphasis bolditalic">3 times the width</em>,” gives us the relationship between the two variables.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p18">Let <em class="emphasis">w</em> represent the width of the rectangle.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p19">Let <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1815" display="inline"><mrow><mn>3</mn><mi>w</mi><mo>−</mo><mn>2</mn></mrow></math></span> represent the length.</p>
<div class="informalfigure large">
<img src="section_04/94c73f76fa94638b53441bfbf09109ca.png">
</div>
<p class="para" id="fwk-redden-ch01_s07_s03_p21">The sentence “<em class="emphasis">A rectangle has a</em> <em class="emphasis bolditalic">perimeter</em> <em class="emphasis">measuring</em> <em class="emphasis bolditalic">92</em> <em class="emphasis">meters</em>” suggests an algebraic set up. Substitute 92 for the perimeter and the expression <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1816" display="inline"><mrow><mn>3</mn><mi>w</mi><mo>−</mo><mn>2</mn></mrow></math></span> for the length into the appropriate formula as follows:</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p22"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1817" display="block"><mtable columnalign="center" columnspacing="0.1em"><mtr><mtd><mi>P</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>2</mn><mi>l</mi></mtd><mtd><mo>+</mo></mtd><mtd><mn>2</mn><mi>w</mi></mtd></mtr><mtr><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd></mtd><mtd><mstyle color="#007fbf"><mtext> </mtext><mo>↓</mo></mstyle></mtd><mtd></mtd><mtd></mtd></mtr><mtr><mtd><mstyle color="#007f3f"><mn>92</mn></mstyle></mtd><mtd><mo>=</mo></mtd><mtd><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>3</mn><mi>w</mi><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mn>2</mn><mi>w</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p23">Once you have set up an algebraic equation with one variable, solve for the width, <em class="emphasis">w</em>.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p24"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1818" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>92</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>w</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>w</mi></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi></mrow><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>92</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn><mi>w</mi><mo>−</mo><mn>4</mn><mo>+</mo><mn>2</mn><mi>w</mi></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>C</mi><mi>o</mi><mi>m</mi><mi>b</mi><mi>i</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>l</mi><mi>i</mi><mi>k</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mi>s</mi></mrow><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>92</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn><mi>w</mi><mo>−</mo><mn>4</mn></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>w</mi></mrow><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>96</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn><mi>w</mi></mtd><mtd></mtd></mtr><mtr><mtd columnalign="right"><mn>12</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>w</mi></mtd><mtd></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p25">Use <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1819" display="inline"><mrow><mn>3</mn><mi>w</mi><mo>−</mo><mn>2</mn></mrow></math></span> to find the length.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p26"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1820" display="block"><mrow><mi>l</mi><mo>=</mo><mn>3</mn><mi>w</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>12</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mo>=</mo><mn>36</mn><mo>−</mo><mn>2</mn><mo>=</mo><mn>34</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p27">To check, make sure the perimeter is 92 meters.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p28"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1821" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>P</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mi>l</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>2</mn><mi>w</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mrow><mo>(</mo><mrow><mn>34</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>12</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>68</mn><mo>+</mo><mn>24</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>92</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p29">Answer: The rectangle measures 12 meters by 34 meters.</p>
</div>
<div class="callout block" id="fwk-redden-ch01_s07_s03_n03">
<h3 class="title">Example 14</h3>
<p class="para" id="fwk-redden-ch01_s07_s03_p30">Given a <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1822" display="inline"><mrow><mn>4</mn><mfrac><mn>3</mn><mn>8</mn></mfrac><mi>%</mi></mrow></math></span> annual interest rate, how long will it take $2,500 to yield $437.50 in simple interest?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p31">Let <em class="emphasis">t</em> represent the time needed to earn $437.50 at <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1823" display="inline"><mrow><mn>4</mn><mfrac><mn>3</mn><mn>8</mn></mfrac><mi>%</mi></mrow><mo>.</mo></math></span> Organize the information needed to use the formula for simple interest, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1824" display="inline"><mrow><mi>I</mi><mo>=</mo><mi>p</mi><mi>r</mi><mi>t</mi></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p32">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="left"><p class="para">Given interest for the time period:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1825" display="inline"><mrow><mi>I</mi><mo>=</mo><mi>$</mi><mn>437.50</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para">Given principal:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1826" display="inline"><mrow><mi>p</mi><mo>=</mo><mi>$</mi><mn>2,500</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para">Given rate:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1827" display="inline"><mrow><mi>r</mi><mo>=</mo><mi> </mi><mn>4</mn><mfrac><mn>3</mn><mn>8</mn></mfrac><mi>%</mi><mo>=</mo><mn>4.375</mn><mi>%</mi><mo>=</mo><mn>0.04375</mn></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch01_s07_s03_p33">Next, substitute all of the known quantities into the formula and then solve for the only unknown, <em class="emphasis">t</em>.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p34"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1828" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>I</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>p</mi><mi>r</mi><mi>t</mi></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007f3f"><mn>437.50</mn></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007f3f"><mn>2500</mn></mstyle><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>0.04375</mn></mstyle></mrow><mo>)</mo></mrow><mi>t</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>437.50</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>109.375</mn><mi>t</mi></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>437.50</mn></mrow><mrow><mstyle color="#007fbf"><mn>109.375</mn></mstyle></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>109.375</mn><mi>t</mi></mrow><mrow><mstyle color="#007fbf"><mn>109.375</mn></mstyle></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>t</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p35">Answer: It takes 4 years for $2,500 invested at <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1829" display="inline"><mrow><mn>4</mn><mfrac><mn>3</mn><mn>8</mn></mfrac><mi>%</mi></mrow></math></span> to earn $437.50 in simple interest.</p>
</div>
<div class="callout block" id="fwk-redden-ch01_s07_s03_n04">
<h3 class="title">Example 15</h3>
<p class="para" id="fwk-redden-ch01_s07_s03_p36">Susan invested her total savings of $12,500 in two accounts earning simple interest. Her mutual fund account earned 7% last year and her CD earned 4.5%. If her total interest for the year was $670, how much was in each account?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p37">The relationship between the two unknowns is that they total $12,500. When a total is involved, a common technique used to avoid two variables is to represent the second unknown as the difference of the total and the first unknown.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p38"> Let <em class="emphasis">x</em> represent the amount invested in the mutual fund.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p39"> Let 12,500 − <em class="emphasis">x</em> represent the remaining amount invested in the CD.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p40"> Organize the data.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p41">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="left"><p class="para">Interest earned in the mutual fund:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1830" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>I</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>p</mi><mi>r</mi><mi>t</mi></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>⋅</mo><mn>0.07</mn><mo>⋅</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007f3f"><mrow><mn>0.07</mn><mi>x</mi></mrow></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para">Interest earned in the CD:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1831" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>I</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>p</mi><mi>r</mi><mi>t</mi></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>12</mn><mo>,</mo><mn>500</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mn>0.045</mn><mo>⋅</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007f3f"><mrow><mn>0.045</mn><mo stretchy="false">(</mo><mn>12</mn><mo>,</mo><mn>500</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
<tr>
<td align="left"><p class="para">Total interest:</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1832" display="inline"><mstyle color="#007f3f"><mrow><mstyle color="#007f3f"><mi>$</mi><mn>670</mn></mstyle></mrow></mstyle></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch01_s07_s03_p42">The total interest is the sum of the interest earned from each account.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p43"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1833" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mi>m</mi><mi>u</mi><mi>t</mi><mi>u</mi><mi>a</mi><mi>l</mi><mi> </mi><mi>f</mi><mi>u</mi><mi>n</mi><mi>d</mi><mi> </mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>t</mi><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>+</mo><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>C</mi><mi>D</mi><mi> </mi><mtext> </mtext><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>t</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mstyle></mtd><mtd><mstyle color="#007fbf"><mo>=</mo></mstyle></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi><mi> </mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>t</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mn>0.07</mn><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>0.045</mn><mrow><mo>(</mo><mrow><mn>12,500</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi> </mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>670</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p44">This equation models the problem with one variable. Solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p45"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1834" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>0.07</mn><mi>x</mi><mo>+</mo><mn>0.045</mn><mrow><mo>(</mo><mrow><mn>12,500</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow><mi> </mi><mi> </mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>670</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0.07</mn><mi>x</mi><mo>+</mo><mn>562.5</mn><mo>−</mo><mn>0.045</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>670</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0.025</mn><mi>x</mi><mo>+</mo><mn>562.5</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>670</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0.025</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>107.5</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>107.5</mn></mrow><mrow><mn>0.025</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4,300</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p46">Use <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1835" display="inline"><mrow><mn>12</mn><mo>,</mo><mn>500</mn><mo>−</mo><mi>x</mi></mrow></math></span> to find the amount in the CD.</p>
<p class="para" id="fwk-redden-ch01_s07_s03_p47"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1836" display="block"><mrow><mn>12,500</mn><mo>−</mo><mi>x</mi><mo>=</mo><mn>12,500</mn><mo>−</mo><mstyle color="#007f3f"><mn>4,300</mn></mstyle><mo>=</mo><mn>8,200</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s07_s03_p48">Answer: Susan invested $4,300 at 7% in a mutual fund and $8,200 at 4.5% in a CD.</p>
</div>
<div class="key_takeaways block" id="fwk-redden-ch01_s07_s03_n05">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch01_s07_s03_l02" mark="bullet">
<li>Solving general linear equations involves isolating the variable, with coefficient 1, on one side of the equal sign. To do this, first use the appropriate equality property of addition or subtraction to isolate the variable term on one side of the equal sign. Next, isolate the variable using the equality property of multiplication or division. Finally, check to verify that your solution solves the original equation.</li>
<li>If solving a linear equation leads to a true statement like 0 = 0, then the equation is an identity and the solution set consists of all real numbers, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1837" display="inline"><mi>ℝ</mi><mo>.</mo></math></span>
</li>
<li>If solving a linear equation leads to a false statement like 0 = 5, then the equation is a contradiction and there is no solution, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1838" display="inline"><mo>Ø</mo><mo>.</mo></math></span>
</li>
<li>Clear fractions by multiplying both sides of an equation by the least common multiple of all the denominators. Distribute and multiply all terms by the LCD to obtain an equivalent equation with integer coefficients.</li>
<li>Simplify the process of solving real-world problems by creating mathematical models that describe the relationship between unknowns. Use algebra to solve the resulting equations.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch01_s07_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd01">
<h3 class="title">Part A: Solving Basic Linear Equations</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch01_s07_qs01_p01"><strong class="emphasis bold">Determine whether or not the given value is a solution.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1839" display="inline"><mrow><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1840" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1841" display="inline"><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>7</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1842" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1843" display="inline"><mrow><mn>3</mn><mi>y</mi><mo>−</mo><mn>4</mn><mo>=</mo><mn>5</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1844" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>9</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1845" display="inline"><mrow><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>7</mn><mo>=</mo><mn>12</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1846" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1847" display="inline"><mrow><mn>3</mn><mi>a</mi><mo>−</mo><mn>6</mn><mo>=</mo><mn>18</mn><mo>−</mo><mi>a</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1848" display="inline"><mrow><mi>a</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1849" display="inline"><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>t</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mo>−</mo><mi>t</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1850" display="inline"><mrow><mi>t</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1851" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>−</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1852" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mi>b</mi><mi>a</mi></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1853" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>2</mn><mi>b</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1854" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mi>b</mi><mi>a</mi></mfrac></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd01_qd02" start="9">
<p class="para" id="fwk-redden-ch01_s07_qs01_p18"><strong class="emphasis bold">Solve.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p19"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1855" display="inline"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>27</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p21"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1856" display="inline"><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>7</mn><mo>=</mo><mn>47</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p23"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1857" display="inline"><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>35</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p25"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1859" display="inline"><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>9</mn><mo>=</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p27"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1861" display="inline"><mrow><mn>9</mn><mi>a</mi><mo>+</mo><mn>10</mn><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p29"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1862" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>3</mn><mi>a</mi><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p31"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1863" display="inline"><mrow><mo>−</mo><mn>8</mn><mi>t</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1865" display="inline"><mrow><mo>−</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>12</mn><mo>=</mo><mn>33</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa17">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1867" display="block"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa18">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1869" display="block"><mrow><mfrac><mn>3</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa19">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1871" display="block"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mn>3</mn><mi>y</mi></mrow><mn>5</mn></mfrac><mo>=</mo><mn>2</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa20">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1872" display="block"><mrow><mfrac><mrow><mn>2</mn><mo>−</mo><mn>5</mn><mi>y</mi></mrow><mn>6</mn></mfrac><mo>=</mo><mo>−</mo><mn>8</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p43"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1873" display="inline"><mrow><mn>7</mn><mo>−</mo><mi>y</mi><mo>=</mo><mn>22</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1874" display="inline"><mrow><mn>6</mn><mo>−</mo><mi>y</mi><mo>=</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p47">Solve for <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1875" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>−</mo><mi>b</mi><mo>=</mo><mi>c</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p49">Solve for <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1877" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd02">
<h3 class="title">Part B: Solving Linear Equations</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd02_qd01" start="25">
<p class="para" id="fwk-redden-ch01_s07_qs01_p51"><strong class="emphasis bold">Solve.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p52"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1879" display="inline"><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>5</mn><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>19</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1880" display="inline"><mrow><mn>7</mn><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1881" display="inline"><mrow><mn>12</mn><mi>x</mi><mo>−</mo><mn>2</mn><mo>−</mo><mn>9</mn><mi>x</mi><mo>=</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1882" display="inline"><mrow><mn>16</mn><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>22</mn><mo>=</mo><mn>8</mn><mo>−</mo><mn>4</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1883" display="inline"><mrow><mn>5</mn><mi>y</mi><mo>−</mo><mn>6</mn><mo>−</mo><mn>9</mn><mi>y</mi><mo>=</mo><mn>3</mn><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p62"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1885" display="inline"><mrow><mn>7</mn><mo>−</mo><mn>9</mn><mi>y</mi><mo>+</mo><mn>12</mn><mo>=</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>11</mn><mo>−</mo><mn>11</mn><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p64"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1886" display="inline"><mrow><mn>3</mn><mo>+</mo><mn>3</mn><mi>a</mi><mo>−</mo><mn>11</mn><mo>=</mo><mn>5</mn><mi>a</mi><mo>−</mo><mn>8</mn><mo>−</mo><mn>2</mn><mi>a</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p66"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1888" display="inline"><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>a</mi><mo>=</mo><mn>5</mn><mi>a</mi><mo>+</mo><mn>7</mn><mo>−</mo><mn>8</mn><mi>a</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p68"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1889" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>=</mo><mfrac><mn>5</mn><mn>6</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p70"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1891" display="inline"><mrow><mfrac><mn>5</mn><mn>8</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>=</mo><mfrac><mn>3</mn><mrow><mn>10</mn></mrow></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p72"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1893" display="inline"><mrow><mn>1.2</mn><mi>x</mi><mo>−</mo><mn>0.5</mn><mo>−</mo><mn>2.6</mn><mi>x</mi><mo>=</mo><mn>2</mn><mo>−</mo><mn>2.4</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p74"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1894" display="inline"><mrow><mn>1.59</mn><mo>−</mo><mn>3.87</mn><mi>x</mi><mo>=</mo><mn>3.48</mn><mo>−</mo><mn>4.1</mn><mi>x</mi><mo>−</mo><mn>0.51</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p76"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1895" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>10</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>8</mn><mo>−</mo><mn>12</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p78"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1896" display="inline"><mrow><mn>8</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo>−</mo><mn>3</mn><mi>x</mi><mo>=</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p80"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1898" display="inline"><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p82"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1899" display="inline"><mrow><mn>7</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>4</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>21</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p84"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1900" display="inline"><mrow><mn>7</mn><mo>−</mo><mn>5</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>t</mi><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>22</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p86"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1901" display="inline"><mrow><mn>10</mn><mo>−</mo><mn>5</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>t</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>20</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p88"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1902" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>4</mn><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p90"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1903" display="inline"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>7</mn><mi>x</mi><mo>=</mo><mn>5</mn><mrow><mo>(</mo><mrow><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-ch01_s07_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p92"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1905" display="inline"><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>4</mn><mi>a</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>5</mn><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p94"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1907" display="inline"><mrow><mn>6</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>b</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>24</mn><mi>b</mi><mo>=</mo><mn>8</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>b</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p96"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1909" display="inline"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>18</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mn>13</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p98"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1910" display="inline"><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p100"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1912" display="inline"><mrow><mn>1.2</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>0.6</mn><mi>x</mi><mo>=</mo><mn>4</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p102"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1913" display="inline"><mrow><mn>6</mn><mo>+</mo><mn>0.5</mn><mrow><mo>(</mo><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2.5</mn><mi>x</mi><mo>+</mo><mn>0.3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p104"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1914" display="inline"><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>15</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>10</mn><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p106"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1916" display="inline"><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>5</mn><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p108"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1917" display="inline"><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>a</mi><mo>+</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1918" display="inline"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>a</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>a</mi></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1920" display="inline"><mrow><mn>6</mn><mo>−</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>7</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>7</mn><mrow><mo>(</mo><mrow><mn>4</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1921" display="inline"><mrow><mn>6</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1923" display="inline"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1925" display="inline"><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mn>4</mn><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1927" display="inline"><mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1929" display="inline"><mrow><mn>6</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>10</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p124">Solve for <em class="emphasis">w</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1931" display="inline"><mrow><mi>P</mi><mo>=</mo><mn>2</mn><mi>l</mi><mo>+</mo><mn>2</mn><mi>w</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p126">Solve for <em class="emphasis">a</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1933" display="inline"><mrow><mi>P</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p128">Solve for <em class="emphasis">t</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1935" display="inline"><mrow><mi>D</mi><mo>=</mo><mi>r</mi><mi>t</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p130">Solve for <em class="emphasis">w</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1937" display="inline"><mrow><mi>V</mi><mo>=</mo><mi>l</mi><mi>w</mi><mi>h</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p132">Solve for <em class="emphasis">b</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1939" display="inline"><mrow><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>b</mi><mi>h</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p134">Solve for <em class="emphasis">a</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1941" display="inline"><mrow><mi>s</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p136">Solve for <em class="emphasis">a</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1943" display="inline"><mrow><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>h</mi><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p138">Solve for <em class="emphasis">h</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1945" display="inline"><mrow><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi mathvariant="italic">π</mi><msup><mi>r</mi><mn>2</mn></msup><mi>h</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p140">Solve for <em class="emphasis">F</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1947" display="inline"><mrow><mi> </mi><mi>C</mi><mo>=</mo><mfrac><mn>5</mn><mn>9</mn></mfrac><mo stretchy="false">(</mo><mi>F</mi><mo>−</mo><mn>32</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p142">Solve for <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1949" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>c</mi></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd03">
<h3 class="title">Part C: Applications</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd03_qd01" start="71">
<p class="para" id="fwk-redden-ch01_s07_qs01_p144"><strong class="emphasis bold">Set up an algebraic equation then solve.</strong></p>
<p class="para" id="fwk-redden-ch01_s07_qs01_p145"><strong class="emphasis bold">Number Problems</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p146">When 3 is subtracted from the sum of a number and 10 the result is 2. Find the number.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p148">The sum of 3 times a number and 12 is equal to 3. Find the number.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p150">Three times the sum of a number and 6 is equal to 5 times the number. Find the number.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p152">Twice the sum of a number and 4 is equal to 3 times the sum of the number and 1. Find the number.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p154">A larger integer is 1 more than 3 times another integer. If the sum of the integers is 57, find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p156">A larger integer is 5 more than twice another integer. If the sum of the integers is 83, find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p158">One integer is 3 less than twice another integer. Find the integers if their sum is 135.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p160">One integer is 10 less than 4 times another integer. Find the integers if their sum is 100.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p162">The sum of three consecutive integers is 339. Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p164">The sum of four consecutive integers is 130. Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p166">The sum of three consecutive even integers is 174. Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p168">The sum of four consecutive even integers is 116. Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p170">The sum of three consecutive odd integers is 81. Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p172">The sum of four consecutive odd integers is 176. Find the integers.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd03_qd02" start="85">
<p class="para" id="fwk-redden-ch01_s07_qs01_p174"><strong class="emphasis bold">Geometry Problems</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p175">The length of a rectangle is 5 centimeters less than twice its width. If the perimeter is 134 centimeters, find the length and width.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p177">The length of a rectangle is 4 centimeters more than 3 times its width. If the perimeter is 64 centimeters, find the length and width.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p179">The width of a rectangle is one-half that of its length. If the perimeter measures 36 inches, find the dimensions of the rectangle.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p181">The width of a rectangle is 4 inches less than its length. If the perimeter measures 72 inches, find the dimensions of the rectangle.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p183">The perimeter of a square is 48 inches. Find the length of each side.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p185">The perimeter of an equilateral triangle is 96 inches. Find the length of each side.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p187">The circumference of a circle measures <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1951" display="inline"><mrow><mn>80</mn><mi mathvariant="italic">π</mi></mrow></math></span> units. Find the radius.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p189">The circumference of a circle measures 25 centimeters. Find the radius rounded off to the nearest hundredth.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd03_qd03" start="93">
<p class="para" id="fwk-redden-ch01_s07_qs01_p191"><strong class="emphasis bold">Simple Interest Problems</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p192">For how many years must $1,000 be invested at <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1952" display="inline"><mrow><mn>5</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>% to earn $165 in simple interest?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p194">For how many years must $20,000 be invested at <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1953" display="inline"><mrow><mn>6</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span>% to earn $3,125 in simple interest?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p196">At what annual interest rate must $6500 be invested for 2 years to yield $1,040 in simple interest?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p198">At what annual interest rate must $5,750 be invested for 1 year to yield $333.50 in simple interest?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p200">If the simple interest earned for 5 years was $1,860 and the annual interest rate was 6%, what was the principal?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p202">If the simple interest earned for 2 years was $543.75 and the annual interest rate was <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1954" display="inline"><mrow><mn>3</mn><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>%</mi></mrow></math></span>, what was the principal?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p204">How many years will it take $600 to double earning simple interest at a 5% annual rate? (Hint: To double, the investment must earn $600 in simple interest.)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p206">How many years will it take $10,000 to double earning simple interest at a 5% annual rate? (Hint: To double, the investment must earn $10,000 in simple interest.)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p208">Jim invested $4,200 in two accounts. One account earns 3% simple interest and the other earns 6%. If the interest after 1 year was $159, how much did he invest in each account?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p210">Jane has her $6,500 savings invested in two accounts. She has part of it in a CD at 5% annual interest and the rest in a savings account that earns 4% annual interest. If the simple interest earned from both accounts is $303 for the year, then how much does she have in each account?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p212">Jose put last year’s bonus of $8,400 into two accounts. He invested part in a CD with 2.5% annual interest and the rest in a money market fund with 1.5% annual interest. His total interest for the year was $198. How much did he invest in each account?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p214">Mary invested her total savings of $3,300 in two accounts. Her mutual fund account earned 6.2% last year and her CD earned 2.4%. If her total interest for the year was $124.80, how much was in each account?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p216">Alice invests money into two accounts, one with 3% annual interest and another with 5% annual interest. She invests 3 times as much in the higher yielding account as she does in the lower yielding account. If her total interest for the year is $126, how much did she invest in each account?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p218">James invested an inheritance in two separate banks. One bank offered <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1955" display="inline"><mrow><mn>5</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>% annual interest rate and the other <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1956" display="inline"><mrow><mn>6</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span>%. He invested twice as much in the higher yielding bank account than he did in the other. If his total simple interest for 1 year was $5,760, then what was the amount of his inheritance?</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s07_qs01_qd03_qd04" start="107">
<p class="para" id="fwk-redden-ch01_s07_qs01_p220"><strong class="emphasis bold">Uniform Motion Problems</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p221">If it takes Jim <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1957" display="inline"><mrow><mn>1</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span> hours to drive the 40 miles to work, then what is Jim’s average speed?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p223">It took Jill <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1958" display="inline"><mrow><mn>3</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> hours to drive the 189 miles home from college. What was her average speed?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s07_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch01_s07_qs01_p225">At what speed should Jim drive if he wishes to travel 176 miles in <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1959" display="inline"><mrow><mn>2</mn><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span> hours?</p>
</div>