# Use Gaussian elimination to find the complete solution to the system of equations, or state that...

## Question:

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

{eq}x+3y+6z=6 \\ y-4z=0 {/eq}

## Gaussian Elimination

Gaussian elimination is the process by which linear equations are solved through performing operations on a matrix representing a set of linear equations. For example, if given 3 equations, they can be represented in matrix form:

{eq}x + y + z = 6 {/eq}

{eq}x + y + 2z = 9 {/eq}

{eq}2x - y = 0 {/eq}

{eq}\begin{matrix} 1 & 1 & 1 & 6 \\ 1 & 1 & 2 & 9 \\ 2 & -1 & 0 & 0 \\ \end{matrix} {/eq}

From there, Gaussian elimination is the process of multiplying entire rows by a coeffient, or adding or subtracting rows from each other to isolate each variable. In the case of the above matrix, the solution is:

{eq}\begin{matrix} 0 & 0 & 1 & 3 \\ 0 & 1 & 0 & 2 \\ 1 & 0 & 0 & 1 \\ \end{matrix} {/eq}

or

{eq}x = 1, y = 2, z = 3 {/eq}

The problem gives only two linear equations:

{eq}x + 3y + 6z = 6 {/eq}

{eq}y - 4z = 0 {/eq}

This can be written in matrix form, where the columns represent the x coefficient, y coefficient, z coefficient, and the constant:

{eq}\begin{matrix} 1 & 3 & 6 & 6 \\ 0 & 1 & -4 & 0 \\ \end{matrix} {/eq}

Because the problem only gives 2 equations, the solution will be in terms of at least one variable. This variable is fairly easy to solve for, as the linear equation {eq}y - 4z = 0 {/eq} can quickly be rearranged to produce {eq}y = 4z {/eq}.

This can be substituted into the first linear equation as:

{eq}x + 3(4z) + 6z = 6 {/eq}

{eq}x + 18z= 6 {/eq}

This produces the following matrix:

{eq}\begin{matrix} 1 & 0 & 18 & 6 \\ 0 & 1 & -4 & 0 \\ \end{matrix} {/eq}

There actually isn't anything left to solve for, as every variable can be isolated in terms of {eq}z {/eq}. In otherwords:

{eq}x = 6 - 18z {/eq}

{eq}y = 4z {/eq}

Given as a solution vector, this can be written as:

{eq}S = \begin{matrix} 6 - 18z \\ 4z \\ z \\ \end{matrix} {/eq} 