# 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.

StepEquationsDescription
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

### References

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

McAdams, David E. Linear System. 4/24/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/l/linearsystem.html.

### 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.)

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