# How do you solve the following system?: -x -4y =31, x -y = -42

Feb 16, 2017

See the entire solution process below:

#### Explanation:

Step 1) Solve the second equation for $x$:

$x - y = - 42$

$x - y + \textcolor{red}{y} = - 42 + \textcolor{red}{y}$

$x - 0 = - 42 + y$

$x = - 42 + y$

Step 2) Substitute $- 42 + y$ for $x$ in the first equation and solve for $y$:

$- x - 4 y = 31$ becomes:

$- \left(- 42 + y\right) - 4 y = 31$

$42 - y - 4 y = 31$

$42 - 5 y = 31$

$- \textcolor{red}{42} + 42 - 5 y = - \textcolor{red}{42} + 31$

$0 - 5 y = - 11$

$- 5 y = - 11$

$\frac{- 5 y}{\textcolor{red}{- 5}} = - \frac{11}{\textcolor{red}{- 5}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 5}}} y}{\cancel{\textcolor{red}{- 5}}} = \frac{11}{5}$

$y = \frac{11}{5}$

Step 3) Substitute $\frac{11}{5}$ for $y$ into the solution for the second equation at the end of Step 1 and calculate $x$:

$x = - 42 + y$ becomes:

$x = - 42 + \frac{11}{5}$

$x = \left(\frac{5}{5} \times - 42\right) + \frac{11}{5}$

$x = - \frac{210}{5} + \frac{11}{5}$

$x = - \frac{199}{5}$

The solution is: $x = - \frac{199}{5}$ and $y = \frac{11}{5}$ or $\left(- \frac{199}{5} , \frac{11}{5}\right)$

Feb 16, 2017

$x = - 39 \frac{4}{5} \mathmr{and} y = 2 \frac{1}{5}$

#### Explanation:

$- x - 4 y = 31 \text{ "and " } x - y = - 42$

This system of equations is a perfect scenario for eliminating
the $x$-terms because they are additive inverses.

$\left(- x\right) + \left(+ x\right) = 0$

$\textcolor{w h i t e}{\ldots \ldots . .} - x - 4 y = 31. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . A$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots . .} x - y = - 42. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots B$

$A + B : \textcolor{w h i t e}{. .} - 5 y = - 11$

$\textcolor{w h i t e}{\ldots . .} y = \frac{- 11}{-} 5 = \frac{11}{5} = 2 \frac{1}{5}$

Substitute $2 \frac{1}{5}$ for $y$ in $B$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots . .} x - 2 \frac{1}{5} = - 42$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots} x = - 42 + 2 \frac{1}{5}$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots} x = - 39 \frac{4}{5}$

NOte that in this case the solutions are easier to work with as mixed numbers rather than as improper fractions which tend to involve large numbers.