Elimination method

System of linear equations can also be solved using the elimination method. We will show how to solve them with some carefully chosen examples. Before you learn this lesson, make sure you understand how to solve linear equations.

Elimination method

Still confused? Here are the steps to follow when solving with the elimination method.

Step 1

Try to eliminate a variable as you add the left sides and the right sides of the two equations.

Step 2 

Set the sum resulting from adding the left sides equal to the sum resulting from adding the right sides.

Step 3

Solve for the variable that was not cancelled or eliminated.

Step 4

Use the answer found in step 3 to solve for the other variable by substituting this value in one of the two equations.

Examples showing how to solve a system of linear equations by elimination using the 4 steps outlined above.


Example #1: Solve the following system using the elimination method.

x + y = 20
x − y = 10

Step 1

Examine the two equations carefully. After a close examination, notice that you can eliminate or cancel the variable y by adding the left sides (x + y and x − y).

However, since you are adding the left sides, you have to add the right sides (20 and 10) of the two equations also to keep the equation balanced.

For the system above, it turns out that it is easy to eliminate y while adding the left sides since x + y + x − y = x + x + y − y = x + x + 0 = 2x.

Step 2

The sum for the left sides is 2x and the sum for the right sides is 20 + 10 = 30

Setting them equal, we get 2x = 30

Step 3

Solve 2x = 30 for x by dividing both sides of this equation by 2.

2x = 30

(2/2)x = 30/2

x = 15

Step 4

You can substitute 15 for x in either x + y = 20 or x − y = 10 to get y.

Choosing the first one, we get 15 + y = 20

Minus 15 from both sides to get y = 5

Now check yourself that the answer is still the same if you had chosen to substitute 15 for x in x − y = 10

Example #2: Solve the following system using the elimination method.

 3x + y = 10
-4x − 2y = 2

Step 1

First, notice that nothing can be eliminated when adding the left sides since
3x + y + -4x − 2y = -1x + -1y

However, in 3x + y = 10, if I can turn y into 2y, I could eliminate the variable y by adding 2y to -2y in -4x − 2y = 2

Therefore, turn y into 2y by multiplying the whole equation by 2

2 ×( 3x + y = 10) gives the new equation 6x + 2y = 20

Adding the left side of 6x + 2y = 20 to the left side of -4x − 2y = 2 gives what you see below:

6x + 2y + -4x − 2y = 6x + -4x + 2y − 2y = 2x + 0 = 2x

Adding the right gives us 20 + 2 = 22

Step 2

The sum for the left sides is 2x and the sum for the right sides is 22

Setting them equal, we get 2x = 22

Step 3

2x = 22

Solve for x by dividing both sides of this equation by 2

2x = 22

(2/2)x = 22/2

x = 11

Step 4

You can substitute 11 for x in either  3x + y = 10 or -4x − 2y = 2 to get y

Choosing the first one, we get 3 × 11 + y = 10

33 + y = 10

Minus 33 from both sides to get y = -23

You should have noticed that the reason we call this method the elimination method is because the first thing you do is eliminate a variable.

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