# How do you solve the system w+4x+3y-11z=42 , 6w+9x+8y-9z=31 and -5w+6x+3y+13z=2, 8w+3x-7y+6z=31?

##### 1 Answer
Feb 20, 2016

$\left(\begin{matrix}w \\ x \\ y \\ z\end{matrix}\right) = \left(\begin{matrix}- \frac{12054}{4889} \\ \frac{38342}{4889} \\ - \frac{31301}{4889} \\ - \frac{14357}{4889}\end{matrix}\right)$

#### Explanation:

Rewrite the equation in linear vector and matrix form:
$\left(\begin{matrix}1 & 4 & 3 & - 11 \\ 6 & 9 & 8 & - 9 \\ - 5 & 6 & 3 & 13 \\ 8 & 3 & - 7 & 6\end{matrix}\right) \left(\begin{matrix}w \\ x \\ y \\ z\end{matrix}\right) = \left(\begin{matrix}42 \\ 31 \\ 2 \\ 31\end{matrix}\right)$
Now use gauss elimination to solve the matrix equation. THe goal here is the convert the 4x4 matrix in half diagonal matrix and solve back from the half diagonal... Used a calculator on Matrix mode to solve:
$\left(\begin{matrix}w \\ x \\ y \\ z\end{matrix}\right) = \left(\begin{matrix}- \frac{12054}{4889} \\ \frac{38342}{4889} \\ - \frac{31301}{4889} \\ - \frac{14357}{4889}\end{matrix}\right)$