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Back-Substitution

The process of solving a linear system of equations that has been transformed into row-echelon form or reduced row-echelon form. The last equation is solved first, then the next-to-last, etc.

 

Example:

Consider a system with the given row-echelon form for its augmented matrix.

The equations for this system are

The last equation says z = 2. Substitute this into the second equation to get

Now substitute z = 2 and y = –13 into the first equation to get

Thus the solution is x = –24, y = –13, and z = 2.

 

See also

Augmented matrix

 

  this page updated 27-aug-12
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons 		NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes