How do you solve 7x+2y=-31 and -5x+y=27 using substitution?

Apr 23, 2017

See the entire solution process below:

Explanation:

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

$- 5 x + y = 27$

$\textcolor{red}{5 x} - 5 x + y = \textcolor{red}{5 x} + 27$

$0 + y = 5 x + 27$

$y = 5 x + 27$

Step 2) Substitute $5 x + 27$ for $y$ in the first equation and solve for $x$:

$7 x + 2 y = - 31$ becomes:

$7 x + 2 \left(5 x + 27\right) = - 31$

$7 x + \left(2 \cdot 5 x\right) + \left(2 \cdot 27\right) = - 31$

$7 x + 10 x + 54 = - 31$

$17 x + 54 = - 31$

$17 x + 54 - \textcolor{red}{54} = - 31 - \textcolor{red}{54}$

$17 x + 0 = - 85$

$17 x = - 85$

$\frac{17 x}{\textcolor{red}{17}} = - \frac{85}{\textcolor{red}{17}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{17}}} x}{\cancel{\textcolor{red}{17}}} = - 5$

$x = - 5$

Step 3) Substitute $- 5$ for $x$ in the solution to the second equation at the end of Step 1 and calculate $y$:

$y = 5 x + 27$ becomes:

$y = \left(5 \cdot - 5\right) + 27$

$y = - 25 + 27$

$y = 2$

The solution is: $x = - 5$ and $y = 2$ or $\left(- 5 , 2\right)$