# How do you solve the following system: 3x - 4y = -23, 6x + 7y = -9 ?

May 7, 2016

$\left(- \frac{3}{2} , \frac{37}{8}\right)$

#### Explanation:

$\left[1\right] 3 x - 4 y = - 23$
$\left[2\right] 6 x + 7 y = - 9$

Get the equivalent of either variable from either equation.
For example, get the equivalent of $x$ from $\left[1\right]$

$3 x - 4 y = - 23$
$\implies 3 x = 4 y - 23$
$\implies x = \frac{4 y - 23}{3}$

Substitute the obtained equivalent of the desired variable into the other equation.

$6 x + 7 y = - 9$

$\implies 6 \left(\frac{4 y - 23}{3}\right) + 7 y = - 9$

$\implies 2 \left(4 y - 23\right) = - 9$

$\implies 8 y - 46 = - 9$

$\implies 8 y = 37$

$\implies y = \frac{37}{8}$

Solve for the other variable now that we know the value of one variable

$3 x - 4 y = - 23$

$\implies 3 x - 4 \left(\frac{37}{8}\right) = - 23$

$\implies 3 x - \frac{37}{2} = - 23$

$\implies 6 x - 37 = - 46$

$\implies 6 x = - 9$

$\implies x = - \frac{9}{6} = - \frac{3}{2}$