# How do you solve using gaussian elimination or gauss-jordan elimination, 2x + 6y = 16, 2x + 3y = -7?

Jun 23, 2016

(x,y)=color(blue)(""(-15,23/3))
(see below for gaus-jordan elimination)

#### Explanation:

Writing the given equations in augmented matrix form:

$\left(\begin{matrix}2 & 6 & 16 \\ 2 & 3 & - 7\end{matrix}\right)$

Pivot element: row:1, column:1
divide all entries in row 1 by 2 in order to reduce the pivot element to 1

$\left(\begin{matrix}1 & 3 & 8 \\ 2 & 3 & - 7\end{matrix}\right)$

subtract 2 times row 1 from row 2 to zero the non-pivot element in the pivot column

$\left(\begin{matrix}1 & 3 & 8 \\ 0 & - 3 & - 23\end{matrix}\right)$

Pivot element: row:2, column:2
divide all entries in row 2 by (-3) in order to reduce the pivot element to 1

$\left(\begin{matrix}1 & 3 & 8 \\ 0 & 1 & \frac{23}{3}\end{matrix}\right)$

subtract 3 times row 2 from row 1 to zero the non-pivot element in the pivot column
(this is the step that converts it from gaussian elimination to gaus-jordan elimination)

$\left(\begin{matrix}1 & 0 & - 15 \\ 0 & 1 & \frac{23}{3}\end{matrix}\right)$

Writing this in non-matrix form:
$\textcolor{w h i t e}{\text{XXX}} x = - 15$
$\textcolor{w h i t e}{\text{XXX}} y = \frac{23}{3}$