Amy will rent a car for the weekend. She can choose one of two plans. The first plan has an initial fee of $50 and costs an additional $0.80 per mile driven. The second plan has no initial fee but costs $0.90 per mile driven. How many miles would Amy need to drive for the two plans to cost the same?

Our two formulas for each plan:

50 (base fee) +.80x (cost per mile) = y (total cost)

.90x (cost per mile) = y (total cost)

We want to find how many miles we have to drive, so solve for x. We're going to plug in y (from equation 2) into the value for y. This is the substitution method.

50+.80x=y

Plugin: 50+.80x =.90x

Add. 90x to each side, subtract 50 from each side:.80x -.90x = - 50

Simplify: -.10x = - 50

Divide: x = - 50/-.10 (two negatives make a positive)

x = 500

So this tells us that she has to drive 500 miles to make the plans equal. Let's verify this by plugging in the value we found for x.

@499 miles, plan 2 is cheaper.

50 +.80 (499) = 449.20

.90 (499) = 449.1

@500 miles, they are equal

50 +.80 (500) = 450

.90 (500) = 450

@501 miles, plan 1 is cheaper

50 +.80 (501) = 450.8

.90 (501) = 450.9

We've verified our answer.