Solving Systems of Equation using the Addition Method (Opposite-Coefficients Method)

This lesson introduces the technique of solving systems of equation using the Addition Method (or Opposite-Coefficients Method).

In some word problems, we may need to translate the sentences into more than one equation. If we have two unknown variables then we would need at least two equations to solve the variable. In general, if we have n unknown variables then we would need at least n equations to solve the variable.

There are two main methods to use when you need to solve more than one equation: Substitution Method and Addition Method.

Check the coefficients of the variables. If the coefficient of one of the variables is 1 then use the Substitution Method otherwise use the Addition Method.

In the Addition Method, the two equations are added together to eliminate one of the variables. We try to get the coefficients of one of the variables to be opposites so that addition will eliminate it.

Let’s look at some examples for this algebra tutorial.

Example 1:

2x + 3y = –2      (equation 1)
4x – 3y = 14      (equation 2)

Solution:

Step 1: In this example the coefficients of y are already opposites (+3 and –3). Just add the two equations to eliminate y.

Step 2: Isolate variable x

6x = 12

Step 3: To get the value of y you need to use the substitution method. Substitute x = 2 into equation 1.

2(2) + 3y = –2
4 + 3y = –2

Step 4: Isolate variable y

3y = –6
y = –2

Step 5: Check your answer with equation 2

4(2) – 3(–2) = 8 – (–6) = 8 + 6 = 14

Answer: x = 2 and y = –2

Example 2:

2x + 3y = 1        (equation 1)
3x – 4y = 10      (equation 2)

Solution:

Step 1: In this example none of the coefficients are opposites. We need to multiply the equations with some numbers to get the coefficients opposite. Lets take the coefficients of y.

Multiply each term of equation 1 by 4

8x + 12y = 4

Multiply each term of equation 2 by 3

9x – 12y = 30

Step 2: Add the two equations to eliminate y.

Step 3: Isolate variable x

17x = 34

Step 4: To get the value of y you need to use the substitution method. Substitute x = 2 into equation 1.

2(2) + 3y = 1

4 + 3y = 1

Step 5: Isolate variable y

3y = –3

y = –1

Step 6: Check your answer with equation 2

3(2) – 4(–1) = 6 – (–4) = 6 + 4 = 10

Answer: x = 2 and y = –1

The following video shows more examples of solving systems of equation using the Addition method.

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