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:

$y+2\\cdot3\\;=\\;-1\\;\\Rightarrow\\;y+6\\;=\\;-1\\;\\Rightarrow\\;y\\;=\\;-7$

Now that the values of y and z are known, these two values can be substituted into the first equation:

$x+3\\cdot(-7)-2\\cdot3\\;=\\;-1\\;\\Rightarrow\\;x-21-6\\;=\\;-1\\;\\Rightarrow\\;x\\;=\\;26$

So the solution to the linear system is:

$x\\;=\\;26,\\quad y\\;=\\;-7,\\quad z\\;=\\;3$

References

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

More Information

Gaussian Elimination and Back Substitution. www.math.usm.edu. Department of Mathematics, University of Southern Mississippi. 6/19/2018. http://www.math.usm.edu/lambers/mat610/sum10/lecture4.pdf.

Cite this article as:

McAdams, David E. Back Substitution. 1/25/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/b/backsubstitution.html.

Image Credits

All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

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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