Explore the Lesson


A. Solving Systems of Linear Equations by Graphing

In previous units, you explored linear equations and used them to solve problems. Sometimes more than one equation, each representing a different scenario, can be written using the same variables. From the Warm Up, the fee structures of the two paintball companies can be represented by linear equations, where C is the total cost and P is the number of paintballs purchased. Equations representing the total cost are:

  • Avenger Paintball: C = 0.1 P + 40

  • Point Zero Paintball: C = 0.15 P + 30


This is an example of a system of equations.

System of Equations
a collection of equations that involve the same variables


Investigation

Tables of values for the cost of using each paintball company are shown.

What do the numbers in each table represent?

Explain how these tables can be used to determine when the number of paintballs purchased and the total cost will be the same for both companies.

When equations include the same variables, they can be plotted on the same grid. The two paintball cost relations are graphed below.

At what point do the two lines intersect? What does this point represent in the paintball scenario?

The point (300, 70) lies on the line represented by C = 0.1 P + 40, but not on the line represented by C = 0.15 P + 30. Substitute 70 for C and 300 for P in each equation. What do you notice?

The point (400, 90) lies on the line represented by C = 0.15 P + 30, but not on the line represented by C = 0.1 P + 40. Substitute 90 for C and 400 for P in each equation. What do you notice?

The point (200, 60) lies on both lines. Substitute 60 for C and 200 for P in each equation. What do you notice?

A solution to a system of equations is a set of variable values that makes each equation true. The point (200, 60) is a solution to the paintball system of equations. This solution can be verified by substituting the coordinates of the point into each equation. If both equations are satisfied, the solution is verified. You may have noticed in the investigation that the point of intersection of the two lines corresponds to the solution. This result can be used to solve systems of equations graphically.

Prior to this unit, solutions to the equations you've worked with have always been single values. Solutions to systems of equations consist of at least one value for every variable in the system.

As such, the solution to a system of equations can be written as an ordered pair, (200, 60), or as separate values, P = 200 and C = 60.

Note that ordered pairs are only used when it is clear which variable is independent and which is dependent.