Systems of Linear Equation in Two Variables: How to Solve by Addition Method

Systems of Linear Equation in Two Variables: How to Solve by Addition Method

Last time I discussed how to solve quadratic equations by factoring . Today, I will teach you how to solve systems of linear equations by ...


Last time I discussed how to solve quadratic equations by factoring. Today, I will teach you how to solve systems of linear equations by addition method. 

Why Solve by Addition Method? 

Solving linear equations with one variable is easy because it only requires addition, subtraction, division, and multiplication.

Linear equation examples:

x + 5 = 3x - 1
2 ( x - 5) = -2
4x - 2 = -3x + 12

But how about solving systems of linear equations in two variables such as 2x - y = 5 and + y = 10? This is when addition method comes into picture. Later, we will solve this system of linear equation in two variables by addition method. 

Solve by Addition Method has five major steps:


1. Look for terms that can be eliminated. If such terms are present, proceed to step 3.
2. Multiply either/both the first and/or second equation by a value such that one variable can be eliminated.
3. Add the two equations and solve for the remaining variable.
4. Substitute the value of the remaining variable in any of the equations to find the value of the eliminated variable.

Solve by Addition Method (Step by Step)

Example 1: Solve the system of equations by addition method.

2x - y = 5
x  + y = 10

1The first step is to look for terms that can be eliminated. In this example, y can be eliminated since -y + y = 0. Following the steps indicated above, we now proceed to step 3.

 2x - y = 5
  x + y = 10

3The third step is to add the two equations and solve for the remaining variable.


 2x - y = 5
 x  + y = 10

 3x      = 15 

 3x = 15     Solve for x.
   x = 5

4The fourth and final step is to substitute the value of the remaining variable in any of the equations to find the value of the eliminated variable.

Tip: Choose the simpler equation. 

In this case, the second equation is simpler than first equation. Hence, we substitute the value of x in the second equation to get the value of y.

Let x = 5

x + y = 10
5 + y = 10 (transpose 5 to the right side of the equation)
y = 5

The solution is (5, 5). 

Example 2: Solve the system of equations by addition method.

x + 2y = 4
3x - y  = 5

1The first step is to look for terms that can be eliminated. In this example, no variable can be eliminated. Therefore, we proceed to step 2.

2The second step is to multiply either/both the first and/or second equation by a value such that one variable can be eliminated. 

You may choose to eliminate either x or y but I personally recommend to eliminate the variable which require a smaller value (absolute) to be eliminated. In this example, we only need to multiply the second equation by 2 so that y can be eliminated rather than multiplying the first equation by -3 to eliminate x.

Note: When I say smaller value, I mean smaller absolute value. Hence, |2| is smaller than |-3|. 

x + 2y = 4
3x - y  = 5     Multiply both sides by 2.

We get:

 x + 2y = 4
6x - 2y = 10

3The third step is to add the two equations and solve for the remaining variable.


 x + 2y = 4
6x - 2y = 10

7x        = 14 

7x = 14     Solve for x.
  x = 2

4The fourth and final step is to substitute the value of the remaining variable in any of the equations to find the value of the eliminated variable. 

Let x = 2

x + 2y = 4
2 + 2y = 4     Transpose 2 to the right side of the equation.
2y = 2           Divide both sides by 2.
y = 1

The solution is (2, 1).

Example 3Solve the system of equations by addition method.

2x + 3y = 7 
3x -  4y = 2  

1The first step is to look for terms that can be eliminated. In this example, no variable can be eliminated. Therefore, we proceed to step 2.

2The second step is to multiply either/both the first and/or second equation by a value such that one variable can be eliminated. 

In this example, we either multiply -3 and 2 or 3 and -2 to the first and second equation, respectively to eliminate x. 

2x + 3y = 7     Multiply both sides by 3. 
3x -  4y = 2     Multiply both sides by -2.

We get:

 6x + 9y = 21
-6x + 8y = -4

3The third step is to add the two equations and solve for the remaining variable.


 6x + 9y = 21
-6x + 8y = -4
       17y = 17

17y = 17     Solve for y.
y = 1

4The fourth and final step is to substitute the value of the remaining variable in any of the equations to find the value of the eliminated variable. 

Let y = 1

-6x + 8y = -4
-6x + 8(1) = -4
-6x + 8 = -4     Transpose 8 on the right side of the equation.
-6x = -12         Divide both sides by -6.
 x = 2

The solution is (2, 1).

If you learned how to solve by addition method after reading this, kindly share this article or leave a comment below. 

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