Applications of Systems of Equations in Two Variables

Applications of Systems of Equations
in Two Variables

The methods for solving systems of equations can be used to solve general problems and applications.

Example

Which is the better value when renting a vehicle?

Rent-A-Car charges \$29.95 per day and 43 cents per mile.
Auto-Loaner charges \$45 per day and 32 cents per mile.

Understand

Find the value at which the costs are the same and then evaluate which plan is most advantageous.

Let m = the total miles to be driven.
Let c = the total rental cost for each company.

Rent-A-Car:     c = 29.95 + 0.43m
Auto-Loaner:    c = 45 + 0.32m

Substitution Method

Since both equations name costs in terms of c, set them equal to each other and solve for miles.  Solving for m will show us when the costs will be the same.

29.95 + 0.43m = 45 + 0.32m
m = 136.82

Rent-A-Car's pricing plan has the higher slope, so from this point onward, its cost will grow at a higher rate than Auto-Loaner's.

Solution

For distances under 137 miles, Rent-A-Car is the better value.
For distances over 137 miles, Auto-Loaner is the better value.