How to Solve a System of Equations by Substitution

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  • 0:02 A System of Equations
  • 1:09 Solving for the First Variable
  • 2:43 Solving for the Rest
  • 3:12 Example
  • 4:27 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can solve a system of equations by using the substitution method. Learn the key to using the substitution method and take a quiz to practice your new skills.

A System of Equations

In this lesson, you'll learn how to solve systems of equations. These are math problems that include more than one equation. On math tests where you have to write things out by hand, the most common systems of equations that you will come across include just two equations, but a system of equations can have as many equations as needed. There are a minimum number of equations required, however: you need as many equations as there are variables. So, if you had four variables in your problem, then you need four equations in order to solve it. There are several methods that you can use for solving, and some of these are discussed in other lessons.

In this lesson, though, you will learn specifically the substitution method. This method involves substituting one equation into another to solve the problem. This method is most easily used on systems of equations with just two equations. When there are more than two equations, this method can get messy on paper very quickly.

Let's take a look at using the substitution method to solve this problem.

x + 3y = 4
x + y = 2

Solving for the First Variable

To begin, you first choose one equation to solve for one of the variables. You can choose any equation and any variable to solve for. Do choose an equation and a variable that is relatively easy to solve for. For example, the second equation looks the easiest to solve for either variable. Go ahead and solve the second equation for x.

x + y = 2
x + y - y = 2 - y
x = 2 - y

If you have three or more equations, you also solve some of the other equations for the other variables. The goal here is to get an equation with just one variable by plugging in the other variables. So look at your equations carefully to see what equations you can use to solve for the variables and which equation you can use to plug those into so that you end up with an equation with just one variable.

Now that you've solved this equation for x, you can now use this information to plug it into the other equation. If you plug in x = 2 - y into the first equation, you will get an equation with just the y variable. By doing this, you will be able to solve for one of the variables, in this case y.

x + 3y = 4
2 - y + 3y = 4
2 + 2y = 4
2 + 2y - 2 = 4 - 2
2y = 2
2y / 2 = 2 / 2
y = 1

Solving for the Rest

Now that you've solved for y, you can use this information and plug it back into your equation x = 2 - y to solve for x.

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