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Solving Linear Systems of Equations by Elimination

A second algebraic method for finding the solution of a system of linear equations is the elimination method.

This method allows us to add two equations to form a new equation. Why add the two equations? In some instances this will result in a new equation that has only one variable. This new equation may then be solved to find the value of that variable. The following example shows how to solve a linear system by elimination.

Note:

The elimination method makes use of the Addition Principle of Equality which states that you can add equivalent quantities to both sides of an equation without changing the solutions of the equation.

 

Procedure — To Solve a Linear System By Elimination

Step 1 Eliminate one variable.

• If necessary, multiply both sides of one or both equations by an appropriate number so that the coefficients of one variable are opposites.

• Add the new equations to form a single equation in one variable.

• Solve the equation.

Step 2 Substitute the value found in Step 1 into either of the original equations and solve.

Step 3 To check the solution, substitute it into each original equation. Then simplify.

 

Example

Use elimination to find the solution of this system.

4x + 3y = 13 First equation

5x - 6y = 52 Second equation

Solution

The coefficients of the x-terms are not opposites.

The coefficients of the y-terms are not opposites.

So, adding the equations will not eliminate a variable.

However, the coefficients of y can be made opposites by multiplying both sides of the first equation by 2.

Step 1 Eliminate one variable.

Multiply both sides of the first equation by 2.

 

2(4x + 3y = 13) 8x + 6y

 

= 26

Add the two equations. Notice the coefficients of y, 6 and -6, are opposites.
8x

5x

+

-

6y

6y

=

=

 

-

26

52

13x + 0y = - 26

Simplify. The y-terms have been eliminated.

Divide both sides by 13.

Now we know x = -2.

Next we will find y.

13x

x

= -26

= -2

Step 2 Substitute the value found in Step 1 into either of the original equations and solve.
We will use the first equation.

Substitute -2 for x.

Multiply.

Add 8 to both sides.

Divide both sides by 3.

The solution is (-2, 7).

 4x + 3y

4(-2) + 3y

-8 + 3y

3y

y

= 13

= 13

= 13

= 21

= 7

Step 3 To check the solution, substitute it into each original equation. Then simplify.

Substitute -2 for x and 7 for y into each original equation and then simplify.

In each case, the result will be a true statement. The details of the check are left to you.

 
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