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.

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=2Original equation
2y=4-x, x-y=2Solve the first equation for y.
3y=4-x, x-(4-x)=2Substitute 4-x for y in the second equation.
4y=4-x, x=3Solve the second equation for x.
5y=4-3, x=3Now substitute the value of x into the first equation.
6y=1, x=3Solve 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. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

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.)

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