How Do I Solve A System of Linear Equations By Graphing?

To solve a system of linear equations in two variables by graphing, you must first solve each equation for the dependent variable. Once this is done, we can use the equations to find an appropriate window for the graph. It is often useful to also solve the system algebraically…this helps us to establish the horizontal extent of the window.

Problem The annual number of cars produced t years after 2000 by a small car manufacturer is $latex y=6.032t+34.543$ thousand cars. A larger producer has annual production $latex y=-10.564t+100.340$ thousand cars. In what year will the annual production be equal? What will the production be then?

Solution Examine the two equations. The vertical intercepts are 34.543 and 100.340. From 34.543, the first line rises. From 100.340, the second line decreases. Based on this, we can deduce that a vertical window from 0 to 110 is appropriate. If we solve the system by substitution, we see that the point of intersection should be t = 3.965. This suggests a horizontal window of 0 to 5 or 0 to 10. In this window, we can find the point of intersection at (3.965, 58.460).

The value 3.965 corresponds to a time late in the year 2003. The number of cars produced at that time is 58.460 thousand cars or 58,460 cars. Be careful in rounding the t value. Although t = 3.965 rounds to 4 (the year 2004), this time is in the year 2003. Rounding to the nearest integer would put the point of intersection in the next year…a mistake when the problem asked in what year.