System of Equations

Systems of equations are a part of the GED Math test.

This lesson is a part of Onsego GED Prep.

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• How to do Ratios

Video Transcription

On the GED Math test, you’ll often encounter questions with two different types of variables, like x and y. You’re already familiar with solving equations with one type of variable, so today, we’ll take those skills to the next level.

When you have questions with two different variables, it’s called solving a system of equations. There are several methods to do this, but the most popular and simplest is the substitution method. This method involves replacing one variable, say x or y, with an algebraic expression.

Let’s look at an example. Suppose you have this equation: x + 2y = 7.
It has two variables, x and y. You’ve learned how to solve equations with one unknown, like x, but now we have two variables. The old method of isolating x won’t work directly here. However, if we know that x equals 5 minus y, we can replace x in the first equation with 5 minus y.
Let’s see how this works. Our original equation x + 2y = 7 becomes (5 minus y) + 2y = 7.
Now, we have only one variable, y, and we can solve this equation using the regular method. Simplify the equation: (5 minus y) + 2y = 7.
Combine like terms to get 5 + y = 7.
Isolate y by subtracting 5 from both sides, and we get y = 2.
Now that we know the value of y is 2, we can plug it into the second equation to find the value of x. So, x equals 5 minus y becomes x equals 5 minus 2, and therefore, x equals 3.
And there you have it! That’s how the substitution method works for solving a system of equations.

Let’s solve a sample quiz question.
What is the solution to the system of equations: 2x + y = 10 and x – y = 2?
A) x = 4, y = 2
B) x = 3, y = 4
C) x = 2, y = 3
D) x = -4, y = 2

To solve this, we’ll use the substitution method. We start with solving one of the equations for one variable and we choose this one x – y = 2.
To isolate x, we add y to both side of the equation, and  we get x = y + 2
Now we’ll substitute x = y + 2 into the other equation:
2 times (y + 2) + y = 10
First, distribute the 2 across the parentheses: 2y + 4 + y = 10
Next, combine like terms: 3y + 4 = 10
To isolate y, we subtract 4 from both sides, and we get 3y = 6
We still need to isolate y, so this time, we divide both sides by 3 and we get y = 2.
Now that we know y equals 2, we can substitute it back into the other equation to find x. We take the equation x = y + 2 and replace y with 2 to get x= 2 + 2 which is 4.
So, the solution to the system of equations is x = 4 and y = 2, which matches Option A.
And that’s how you solve a system of equations using the substitution method.

Last Updated on January 15, 2026.