# How can matrices be used when solving by substitution?

Apr 7, 2018

The process of "solving" a matrix and solving a system of equations are mathematically identical.

Suppose you have some system with $3$ unknowns and $3$ equations. Say,

$3 x + 4 y + 9 z = 26$,
$5 x - 2 y + 3 z = 13$,
$x + y - 2 z = 10$.

This is equivalent to the matrix equation

$\left(\begin{matrix}2 & 4 & 9 \\ 5 & - 2 & 3 \\ 1 & 1 & - 2\end{matrix}\right) \cdot \left(\begin{matrix}x \\ y \\ z\end{matrix}\right) = \left(\begin{matrix}26 \\ 13 \\ 10\end{matrix}\right)$.

To solve this, one would set up an augmented matrix and use row-reduction techniques to find your solution. The bolded methods are outside the scope of basic algebra and you will run across them if you ever take a linear algebra course.

Just know that row-reduction techniques simulate substitution and that it is not (mathematically) easier to use a matrix to solve a system than to solve a system in the traditional way.