# How do you solve the following system:  8x-7y=-39, 5x-9y=-23 ?

Jul 4, 2017

$x = - \frac{190}{37} \mathmr{and} y = - \frac{11}{37}$

Be sure to work with fractions, not decimals which will be recurring and will require rounding, leading to inaccurate answers.

#### Explanation:

The ideal scenario is to have the numerical coefficients of one of the variables as additive inverses.

$\text{ "color(blue)(8)x-7y =-39" } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . A$
$\text{ "color(red)(5)x-9y=-23" } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . B$

$A \times 5 \text{ "color(blue)(40)x-35y =-195" } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . C$
$B \times \text{-"8" "color(red)("-"40)x+72y =+184" } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . D$

$C + D : \text{ "37y = -11" } \leftarrow x$ is eliminated

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} y = - \frac{11}{37}$

Now substitute the value for $y$ into one of the equations

In $A : \text{ } 8 x - 7 \left(- \frac{11}{37}\right) = - 39$

$8 x + \frac{77}{37} = - 39$

$8 x = - 39 - \frac{77}{37}$

$8 x = - 39 - 2 \frac{3}{37}$

$8 x = - 41 \frac{3}{37}$

$8 x = - \frac{1520}{37}$

$x = - \frac{1520}{8 \times 37}$

$x = - \frac{190}{37}$

Who comes up with these equations???