Solving 3x3 Systems of Linear Equations

Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

Have you ever solved an equation with three variables in it? In this lesson, you will learn how to solve systems of equations that have three equations and three variables.

What Is a 3x3 System of Linear Equations?

To define a 3x3 system of linear equations we need to understand what each part of the term means. Linear equations form straight lines when they are graphed. They have a degree of one, meaning that the variables have an exponent no greater than one. A system of equations has two or more equations that are solved simultaneously. When a system of equations is 3x3, it has three equations and three variables. The goal of solving a system of equations is to find a value for each of the variables that satisfies all of the equations. In a 3x3 system of linear equations, we need to find a value for each of the three variables that makes each equation true.

Solving with the Substitution Method

In the example, we see how an expression from one equation can be substituted for a variable in another equation. The goal is to have an equation with one variable that we can solve for. Once we find the value of one variable, we can use it to solve for the others.

Example of a 3x3 system of linear equations

The first equation tells us that x is equal to the expression 2y + 1. We can substitute this expression for x in the other equations. The second equation becomes z = -3(2y + 1) = -6y - 3. The expression -6y - 3 can be substituted for z in the third equation. After substituting the expressions for x and z, the third equation becomes (2y + 1) + 3y - (-6y - 3) = 4.

Applying the substitution method

Now that we have one equation with just one variable, we can solve for the variable y.

Solving for the variable y

Since y = 0, we can substitute it into the other equations and solve for the other variables. For the first equation, we get x = 2(0) + 1 = 1. Substituting one for x in the second equation gives us z = -3(1) = -3. We have now found the values for all three variables.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account