Dynamic Algebra™ Exploration

The Eliminator

When you add equal quantities to each side of an equation, the resulting equation has the same solution as the original. In the same way, when you add two quantities that are equal, to two other quantities that are equal, the resulting expressions are equal.

The elimination method makes use of this fact to solve systems of linear equations. This page will help you understand the elimination method by looking at the graphs of the linear equations in the system. The sketch can help with Lesson 5.3 of Discovering Algebra: An Investigative Approach, especially with Exercise 12 on page 294.

Sketch

This sketch allows you to add multiples of two equations and see the result. You can input the original two equations by adjusting sliders A, B, and C for the blue equation and sliders D, E, and F for the green equation. These sliders are aligned with the x-axis. The graphs of the equations are shown in matching colors.

Press Show Multipliers to see the sliders S and T. You can adjust slider S to create a multiple of the blue equation, and slider T to create a multiple of the green equation.

Press Show Sum to see the equation and graph that result from adding the multiples of the original equations. In other words, the equation and graph in red are the sum of S times the blue equation Ax + By = C and T times the green equation Dx + Ey = F.

Investigate

1. Set the sliders A, B, C, D, E, and F to give the two equations of the system you want to solve. You may be able to estimate the solution by looking at the intersection of the two lines.
2. Press Show Sum to show the sum of the equations and the corresponding graph, both in red. What do you notice?
3. Press Show Multipliers. Drag the sliders S and T until the red line is either horizontal or vertical. You may have to adjust one or both sliders. Alternatively, you might wish to use the elimination method to solve the system algebraically, then adjust the sliders to match the values you used to multiply each equation.
4. Read off one part of the solution from the red equation. Substitute this solution back into one of the original equations to find the other part of the solution. Then change the values of sliders S and T to show this other part of the solution as a vertical or horizontal line.
5. What part of the solution can you get from a horizontal line? Why?
6. What part of the solution can you get from a vertical line? Why?
7. When does a system of two linear equations have no solution? In this case, what can you say about the red line? Why is this true?