Back Substitution

Pronunciation: /bæk ˈsʌb.stɪˌtut.ʃən/ Explain

Back substitution is an algorithm for solving a linear system that has been reduced to row-eschelon form, which is an upper triangular matrix:

\\left[\\array{1 & a_{12} & a_{13} & a_{14} \\\\ 0 & 1 & a_{23} & a_{24} \\\\ 0 & 0 & 1 & a_{34}}\\right]
Since the rows in a matrix represent equations, and the columns represent variables in those equations, the matrix
\\left[\\array{1 & 3 & -2 & -1 \\\\ 0 & 1 & 2 & -1 \\\\ 0 & 0 & 1 & 3}\\right]
can be expressed as the linear system
\\array{ x+3y-2x=-1 \\\\ y+2z=-1 \\\\ z=3 }
The value of z is known, so 3 can be substituted into the equation just above it:
Now that the values of y and z are known, these two values can be substituted into the first equation:
So the solution to the linear system is:
x\\;=\\;26,\\quad y\\;=\\;-7,\\quad z\\;=\\;3


  1. McAdams, David E.. All Math Words Dictionary, back substitution. 2nd Classroom edition 20150108-4799968. pg 23. Life is a Story Problem LLC. January 8, 2015. Buy the book

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Cite this article as:

McAdams, David E. Back Substitution. 1/25/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

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12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/19/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
11/20/2008: Initial version. (McAdams, David E.)

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