Linear System

Pronunciation: /ˈlɪn.i.ər ˈsɪs.təm/ Explain

A linear system is a set of lines or linear objects that are all simultaneously true (true given the same circumstances). Together they define the solution(s) to the system.

Two methods commonly used to solve a linear system are:

  • substitution and elimination; and
  • Gauss-Jordan elimination.

How to Solve a Linear System by Substitution and Elimination

Substitution and elimination is a method for solving linear system. Given two or more linear equations, this method may be used to find the solution the linear system.

1x + y = 4
x - y = 2
Original equation
2y = 4 - x
x - y = 2
Solve the first equation for y.
3y = 4 - x
x - (4 - x) = 2
Substitute 4 - x for y in the second equation.
4y = 4 - x
x = 3
Solve the second equation for x.
5y = 4 - 3
x = 3
Now substitute the value of x into the first equation.
6y = 1
x = 3
Solve for y. The solution to this linear system is y = 1, x = 3.
Example 1


  1. McAdams, David E.. All Math Words Dictionary, linear system. 2nd Classroom edition 20150108-4799968. pg 109. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Linear System. 4/24/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

Revision History

4/24/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/31/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
11/27/2008: Changed equations to images. (McAdams, David E.)
8/23/2008: Added substitution and elimination example. (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2018 Life is a Story Problem LLC. All rights reserved.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License