# Create a set of system of linear equations to answer the following.

A factory is currently running at $85\%$ of its original capacity, and management is considering upgrading the equipment. The upgrade will take $6$ months, during which time the factory will not run at all. Once complete, the factory’s output will increase to $120\%$ of the original capacity.

After how long would the upgraded factory’s production match the current $85\%$ production? In other words, how long will it take for the factory to make up for the loss of $6$ months?

This what I have come up with so far:

$0.85E - 6m = 100$ and $1.20E + 6m = m$

$E$ represents the percentage capacity, $m$ represents the months, and $100$ represents the factory original output of $100$ units per month.

I understand how to solve using system of equations but I am having difficult with coming up with a set of equations for this problem in particular.

## 1 Answer

So after 6 months, with the new equipment, the factory will produce at 120 units per month, which is 120-85 = 35 units better, so how long does it take for 35 units more a month to be worth 6 months at 85 units?

$$6*85 = 35 * t$$

$$t = 510/35 = 14.571$$

So after 1 year and 3 months will have completely recovered its losses