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.