Answer to Question #343003 in Linear Algebra for Miki

Question

A company makes products A and B. Each unit of product A requires 1 unit of resource 1 and 5 units

of resource 2. Each unit of product B requires 5 units of resource 1 and 4 units of resource 2. If there

are 40 units of resource 1 and 140 units of resource 2 available, how many units of each product should

be produced if all the resources are to be used?

Answer 1

Let $a$ be the number of units of product A produced, $b$ be the number of units of product B produced.

Then

$a\cdot1+b\cdot5=40\\ a\cdot5+b\cdot4=140.$

We have a system of linear equations:

$a+5b=40\\ 5a+4b=140.$

First, we will solve the first equation for $a$:

$a=40-5b.$

Now we can substitute the expression $40-5b$ for $a$ in the second equation.

$5(40-5b)+4b=140\\ 200-25b+4b=140\\ 21b=60\\ b=\cfrac{60}{21}=\cfrac{20}{7}\approx2.86.$

Now, we substitute $b=\cfrac{20}{7}$ into the first equation and solve for $a$:

$a=40-5\cdot \cfrac{20}{7}=\cfrac{40\cdot7-5\cdot20}{7}=\cfrac{180}{7}\approx25.71.$

So, 25.71 units of product A and 2.86 units of product B should be produced if all the resources are to be used.