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Let us Understand the Least Understood, Fair and Square! |
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We encounter equations very often in our lives. Consider for example, the following situation at Baker's Cafe. The manager has a very important estimate to make. Mostly, visitors at his cafe happen to be families and they are often comprised of Children and/or Adults. He observes that there are 3 adults and 1 child at a table and their bill turns out to be Rs.1200/-. There is yet another table with 2 children and 1 adult and their bill comes out to be Rs.1000/-. Can the manager estimate the consumption of a Child/Adult? This is popularly called the Simultaneous Equations and we all remember from our school days, multiple ways in which these can be solved.
$$ 3A + 1C = 1200 $$
$$ 1A + 2C = 1000 $$ -
While we were taught the so called two variables and two unknowns, what if there were more equations than unknowns?
$$ 3A + 1C = 1200 $$
$$ 1A + 2C = 1000 $$
$$ 1A + 1C = 900 $$ -
Note that the previous question can be modelled as a matrix:
$$ 3A + 1C = 1200 $$
$$ 1A + 2C = 1000 $$
$$ 1A + 1C = 900 $$Observe this is same as :
$$ \left( \begin{matrix} 3 & 1 \1 & 2 \1 & 1 \\end{matrix}\right) $$ $$\left( \begin{matrix} B\ C\ \end{matrix} \right)$$
$$ \left( \begin{matrix} 1200\ 1000\ 900\ \end{matrix} \right) $$
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The best way to solve is, is to guess the values :-). Can you write a python code to guess the values?
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How do you solve this mathematically? There are two nice ways of solving this:
- Model this as a question of inverting a rectangular matrix and find the solution. (Whoa! my teacher never taught me that!)
- Model this as a funtion of two variables and solve it using partial differentiation (Eeks!)