System of Linear Equations: Definition & Examples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: What is an Inequality?

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:04 Linear Equations
  • 1:23 System of Linear Equations
  • 2:19 System of Linear…
  • 3:32 Practice Problem
  • 4:30 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

A system of linear equations can come in handy when we come across problems where we have more than one quantity to find. In this lesson, we'll learn how to define a system of linear equations and look at examples.

Linear Equations

Imagine telling a friend about your favorite coffee and doughnut shop. Your friend asks how much it charges for a cup of coffee and a doughnut. While you can't remember how much each one costs individually, you do remember that the last time you ordered one coffee and two doughnuts, your bill was $7.00. And today, when you bought two coffees and three doughnuts, your bill came to $12.00.

Believe it or not, this is all the information you need to answer your friend's question. To do this, we use what is called a system of linear equations. A linear equation is a polynomial equation in which the unknown variables have a degree of one. That is, all of the unknown variables in a linear equation are raised to the power of one. Some examples of a linear equation are shown in the image below.

system of linear equations 1

These equations are polynomial equations in which the variables are raised to the power of one. A couple of non-examples are shown in this other image below.

system of linear equations 2

In the equation, 2x^2 + 3y - 4 = 0, the variable x is raised to the power of 2, so this is not a linear equation. The equation, y = 3x / (x - 1), isn't a linear equation because, while its variables are raised to the power of one, it isn't a polynomial equation.

System of Linear Equations

A system of linear equations is a set of two or more linear equations with the same variables. For example, the sets in the image below are systems of linear equations.

system of linear equations 3

Let's return to the question your friend asked about the cost of a cup of coffee and a doughnut at your favorite coffee shop. Here, c represents the cost of one coffee, and d represents the cost of one doughnut. The linear equation for your purchase of one coffee and two doughnuts, which came to $7.00, can be written as follows: c + 2d = 7. Similarly, your purchase of two coffees and three doughnuts, for a total of $12.00, can be expressed as: 2c + 3d = 12.

Now, let's put these two equations together. In partnership, they give us the system of linear equations required to figure out the cost of one coffee and the cost of one doughnut.

2c + 3d = 12

c + 2d = 7

System of Linear Equations Solution

A solution to a system of linear equations is a set of numbers that, when we substitute numbers for specified variables in the system, makes each equation in the system a true statement. For example, if we plug 4 in for x and 7 in for y, both of the equations in the following system are true statements.

x - y = -3

x + y = 11

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