Solving Graphically

Kim Seay

EMAT 6680

Begin with any equation. I will use 3x-4=11. You can graph both sides of the equation on the same plane by graphing y=3x-4 and y=11.

The point where the two lines intersect (which is identified by the square above) is the solution of the equations. You can see from above that the point of intersection is (5, 11), and 3(5)-4=11.

Car Rental Quandary

To show students how this type of information might be useful in every day life, we can create a scenario involving two car rental stores with different rates for customers.

Suppose store A charges an initial fee of $50.00 to rent a car plus an additional $.20 per mile. The charge of this store per mile could be seen by the graph of y=50+.20x where x= the number of miles.

Store B charges an initial rental fee of $100.00, but only $.05 per mile. The charge of this store could be seen by graphing the equation y=100+.05x where x=number of miles driven.

The question we want to answer, is when is the final rate charged by the two stores equal? In other words, for what value of "x" will 50+.20x=100+.05x? We can solve this by graphing the two equations like we did above.

y = 50 + .20x

y=100 + .05x

The two graphs intersect approximately at the point (333.33, 116.67). This tells us that if we drive 333 and one third miles, both car rental stores will charge $116.67.


This is something that would be very easy to demonstrate in the classroom. I think this exploration would be just as effective with a graphing calculator , and perhaps more of the students would be able to use that knowledge to help them solve equations later.