Session 6, Part D:
Solving Systems of Equations (30 minutes)

 In This Part: Solving Balance Equations by Graphing | Non-Linear Graphs For Part D, we will need 14 counters, seven each of two colors, and some tape. Coins may be used in place of colored counters. In Part C, we encountered some balance problems that were not easily represented with bags and blocks. Here is Problem C7: Can you draw a balance puzzle to represent the equation 4x - 2 = 5x - 3? Note 10 Using bags and blocks for Problem C7, for example, would have required that we draw "negative" blocks — not an easy task! Another way to solve the equation in Problem C7 is to think about each side of the equation as a function. Think of creating two functions, one to represent each side of the equation. We'll use the variable y to represent the total for each side of the equation. Using x and y as variables, the left side of the equation in Problem C7 corresponds to the line equation y = 4x - 2. The right side of the equation corresponds to the line equation y = 5x - 3. To solve the original equation, we want to find common solutions to both of the new equations. A common solution is called a solution to a system of equations. In this section, we will take a closer look at solving systems of linear equations by graphing. Graphs are useful tools for visualizing how to solve equations. We will create our own coordinate system, using tape to represent the axes and counters to represent the points of each graph. First, we will need to create the axes. The x-axis is horizontal, the y-axis is vertical, and the two should intersect in the center of the area we're using. The x-axis should be labeled in increments from -3 to +3, and the y-axis should be labeled in increments from -10 to +10. If you prefer, you may print out this picture of the coordinate system: Seven counters will make the first graph. Follow these rules:
 a. Line up the counters along the x-axis at each integer value between and including -3 and +3. b. At each point, multiply the x value by 2, then subtract 3 from the result. Move the counter vertically to the final result of this calculation, making sure not to move horizontally.

 Problem D1 Describe the resulting graph.

 Problem D2 What is its equation?

 Refer back to Session 5 on line equations if you need a refresher.    Close Tip

 Problem D3 Use two points on the graph to describe the slope. Count the rise and run from one point to the other. Note 11

The next graph requires two sets of five counters. For the first graph, follow these instructions:

 a. Line up the counters along the x-axis at each integer value between and including -2 and +2. b. At each point, multiply the x value by 2, then subtract 1 from the result. Move the counter vertically to the final result of this calculation, making sure not to move horizontally.

Now, on the same axes, follow these instructions for the second graph:

 a. Line up the counters along the x-axis at each integer value between and including -2 and +2. b. At each point, multiply the x value by -1, then add 5 to the result. Move the counter vertically to the result of this calculation.

 Problem D4 Do these two graphs share a point?

 Problem D5 What is the significance of this point?

 The points on each line are the solutions to the equation for that line. What is true of a point which is shared by both lines?   Close Tip

 Problem D6 Use the same method to find a solution to the system y = -3x and y = x - 4.

 Problem D7 Use counters to find the solution to Problem C7 below: Can you draw a balance puzzle to represent the equation 4x - 2 = 5x - 3?

 Turn each side of the equation into an equation for a line. Then graph the lines by using each line's equation to find counter-moving rules.    Close Tip

 Session 6: Index | Notes | Solutions | Video