# How do you solve the system -2x + 2y= -5 and x + y= -5?

Mar 18, 2018

$x = - \frac{5}{4}$, $\text{ } y = - \frac{15}{4}$

#### Explanation:

Eq1: $- 2 x + 2 y = - 5$
Eq2: $x + y = - 5$

We need to find one of the variables. Since Eq2 already has variables with coefficients of $1$, let's start there. Let's solve for $x$ in Eq2:

Eq3: $x = - y - 5$

Now we can use Eq3 by substituting it into Eq1. We cannot substitute into Eq2, as we have already used this equation to derive a result!

Eq3 -> Eq1:
$- 2 \left(- y - 5\right) + 2 y = - 5$
$\implies 2 y + 10 + 2 y = - 5$
$\implies 4 y + 10 = - 5$
$\implies 4 y = - 15$
$\implies \textcolor{b l u e}{y = - \frac{15}{4}}$

We now have a value for $y$. We have both Eq1 and Eq2 that must hold. We can use whichever one, as both will give the same results. Let's just substitute into Eq2 for convenience:

$x + \left(\textcolor{b l u e}{- \frac{15}{4}}\right) = - 5$
$\implies \textcolor{\mathmr{and} a n \ge}{x = - \frac{5}{4}}$

Now we have values for both $x$ and $y$:
$x = - \frac{5}{4}$,$\text{ } y = - \frac{15}{4}$