Given a system of two algebraic equations with two unknown variables there are two main methods to solve for the unknown values. An example of such a system of equations would be:
x + y = 5
x – y = 1
OR
2x + y = 9
y – x = 3
The first method for solving this system of equations is called substitution.
When we apply the method of substitution there are five main steps which we need to remember.
Step 1: Label your equations 1 and 2.
Step 2: Begin with equation 1 and isolate one variable.
Step 3: Substitute equation 1 into equation 2.
Step 4: Solve for your first variable.
Step 5: Substitute the value of your first variable into one of the equations and solve for the second unknown variable
Now, we will try an example.
Example 1: x + y = 8 and x – y = 4
Step 1: Label x + y = 8 as equation 1. Label x – y = 4 as equation two.
Step Two: Consider equation 1 and isolate for the variable x.
x + y = 8
x + y – y = 8 – y
x = 8 – y
Step Three: Substitute x = 8 – y into equation 2. Once we do so, equation 2 will become:
x – y = 4
(8 – y) – y = 4
Step Four: Solve for your first variable.
8 – 2y = 4
8 – 2y – 8 = 4 – 8
-2y = -4
2y = 4
y = 2
Step Five: Let y = 2 in equation 1 and solve for x.
x + y = 8
x + 2 = 8
x + 2 – 2 = 8 – 2
x = 8 – 2
x = 6
Therefore, we have found that the solution to this system of equations is y = 2 and x = 6.
The second method for solving this system of equations is called elimination.
If we use the method of elimination in order to solve a system of equations there are four important steps.
Step One: Label your equations 1 and 2.
Step Two: Add or subtract a multiple of equation 1 with equation 2 so that one of the unknown variables is removed.
Step Three: Solve for the first unknown variable.
Step Four: Substitute the value of the first variable into one of the equations and solve for the second unknown variable.
We will now, walk through the steps in an example.
Example 2: x + y = 10 and x – y = 4
Step One: Label x + y = 10 as equation 1 and label x – y = 4 as equation 2.
Step Two: Perform the operation of equation 1 plus equation 2.
x + y = 10
+ x – y = 4
2x = 14
Note: We choose this operation just when we add equation 1 and equation 2 the variable y cancels out and only one variable, x, remains in the answer. This is what we want. When choosing which operation to use look for how you can eliminate one of the unknown variables while the other variable remains in the answer.
Step Three: Solve for the first unknown variable.
2x = 14
x = 7
Step Four: Substitute x = 7 into equation 1 and solve for the value of y.
7 + y = 10
7 + y – 7 = 10 – 7
y = 10 – 7
y = 3
Therefore, we have now found that the solution to this system of equations is x = 7 and y = 3.
Now, you give it a try.
Can you show that the solution to the system of equations, x + y = 15 and y – 2x = 0, is x = 5 and y = 10?
Can you find the solution to the system of equations, x + 2y = 7 and x + y = 5?
This article was written for you by Mia, one of the tutors with Test Prep Academy.