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# Elimination Using Addition and Subtraction

Objective Introduce the elimination method of solving systems of simultaneous equations.

The main idea in this lesson is that systems of equations may be simplified and solved using the Addition and Multiplication Properties of Equality to add and to subtract equations. The goal is to obtain an equivalent equation in which only one variable is on the left-hand side and only a number is on the right-hand side.

## The Elimination Method for Solving Systems of Equations

Let's start with some examples.

Example 1

Solve the system of linear equations.

x - y = 3

x + y = 7

Solution

For (x , y ) to be a solution to the system means that both equations hold true for the values (x , y ). That is, both equations are satisfied. Add the equations to get another valid equation, using the Addition Property of Equality.

 x - y = 3 (+) x + y = 7 Add the equations. 2x + 0 = 10 2x = 10

This is a linear equation in one variable, namely x. Solve to get x = 5. Then substitute x = 5 into one of the original equations and solve for y.

 x - y = 3 5 - y = 3 Replace x with 5. -y = -2 y = 2

The only solution to the system of equations is (5, 2).

Example 2

Solve the system of equations.

2x + 5y = 7

2x - 2 y = 0

Solution

Use the Subtraction Property of Equality to subtract one of the equations from the other to get another valid equation.

 2x + 5y = 7 (-) 2x - 2 y = 0 Subtract the equations. 0 + 7y = 7

Solve 7y = 7 to get y = 1. Then substitute the value of y into one of the equations and solve for x.

 2x - 2(1) = 0 Substitute 1 for y. 2x - 2 = 0 2x = 2 x = 1 Divide each side by 2.

The solution is (1, 1).

When the resulting equation involves only one variable, we say that we have eliminated the other variable. In Example 1, y was eliminated, and in Example 2, x was eliminated. Note to the Teacher It is important for students to practice using this method by working the exercises below. Then, introduce Examples 3 and 4, which are word problems.

Example 3

The sum of two numbers is 48, and their difference is 16. What are the numbers?

Solution

Let x and y represent the two numbers and write the following system of equations.

x + y = 48

x - y = 16

Adding the equations gives 2x = 64 or x = 32. To find the value of y, substitute 32 for x in either equation to get y = 16. The two numbers are 32 and 16.

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