Answer to Question #343003 in Linear Algebra for Miki

Question #343003

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?

1
Expert's answer
2022-05-22T23:37:14-0400

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\\\\\na\\cdot5+b\\cdot4=140."

We have a system of linear equations:

"a+5b=40\\\\\n5a+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\\\\\n200-25b+4b=140\\\\\n21b=60\\\\\nb=\\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.


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