# How do you solve the following system?: -13x -3y =6, 8x -y = -1

Nov 19, 2015

$\left(x , y\right) = \left(- \frac{9}{37} , \frac{35}{37}\right)$
(see below for method of solution)

#### Explanation:

Given
[1]$\textcolor{w h i t e}{\text{XXX}} - 13 x - 3 y = 6$
[2]$\textcolor{w h i t e}{\text{XXX}} 8 x - y = - 1$

Multiply [2] by 3 (to give $y$ the same coefficient as in [1])
[3]$\textcolor{w h i t e}{\text{XXX}} 24 x - 3 y = - 3$

Subtract [1] from [3] (to get rid of the $y$ term)
[4]$\textcolor{w h i t e}{\text{XXX}} 37 x = - 9$

Divide [4] by 27
[5]$\textcolor{w h i t e}{\text{XXX}} x = - \frac{9}{37}$

Re-write [2] as an equation for $y$ by adding $\left(y + 1\right)$ to both sides and switching the sides
[6]$\textcolor{w h i t e}{\text{XXX}} y = 8 x + 1$

Substitute $\left(\frac{9}{37}\right)$ from [5] for $x$ in [6]
[7]$\textcolor{w h i t e}{\text{XXX}} y = 8 \left(- \frac{9}{37}\right) + 1$

Simplify [7]:
[8]$\textcolor{w h i t e}{\text{XXX}} y = - \frac{35}{37}$

Probably the biggest problem here is trusting your calculations (or being willing to do them in the first place) since the answer isn't "pretty".

If it helps a graph can help verify that the answer is within an appropriate range:
graph{(-13x-3y-6)*(8x-y+1)=0 [-3.97, 3.826, -2.976, 0.917]}