# How do you solve the following system using substitution?: 5x – 2y = -5, y – 5x = 3

May 28, 2018

$x = \left(- \frac{1}{5}\right)$
$y = 2$

#### Explanation:

$5 x - 2 y = - 5$
$y - 5 x = 3$

Solving by Substitution

First, you want to find the equation for a variable that you can replace in the system. $y - 5 x = 3$ is an equation that appears easy to re-arrange to get an equation for a variable, so we'll use it:

$y - 5 x = 3$

Add $5 x$ to both sides to cancel out $- 5 x$ in order to get the equation for the value of y. You should now have:

$y = 5 x + 3$

Now that you have an equation for a variable, substitute these terms ($5 x + 3$) in the first equation of the system. So:

$5 x - 2 y = - 5$ becomes
$5 x - 2 \left(5 x + 3\right) = - 5$.

Distribute $- 2$ to the terms inside the parentheses. You do this by multiplying $- 2$ by each term, so:

$- 2 \cdot 5 x = - 10 x$
$- 2 \cdot 3 = - 6$

Re-write your equation to reflect new information:

$5 x - 10 x - 6 = - 5$

Combine like terms.

$- 5 x - 6 = - 5$

Add $6$ to both sides to cancel out $- 6$. You should now have:

$- 5 x = 1$

Divide by $- 5 \to i s o l a t e f \mathmr{and}$x#. You should now have:

$x = - \frac{1}{5}$

Plug the value of $x$ into the equation for the value of $y$:

$y = 5 x + 3$
$y = 5 \left(- \frac{1}{5}\right) + 3$
$y = - 1 + 3$
$y = 2$

Plug these values back in to confirm that they're right:
$5 x - 2 y = - 5$
$5 \left(- \frac{1}{5}\right) - 2 \left(2\right) = - 5$
$- 1 - 4 = - 5$
$- 5 = - 5$

$y - 5 x = 3$
$2 - 5 \left(- \frac{1}{5}\right) = 3$
$2 - - 1 = 3$
$2 + 1 = 3$
$3 = 3$

These values are correct.