# How do you solve the following system using substitution?: 3(x+5) - 2(y - 1) = 6, x + 4(y + 3) = 13

Nov 1, 2015

$x = - 3$, $y = 1$

#### Explanation:

To solve the system using substitution, first we need to transform one of the equations such that one of the variables is expressed as an equality in terms of the other variable(s).

$\left[1\right] 3 \left(x + 5\right) - 2 \left(y - 1\right) = 6$
$\left[2\right] x + 4 \left(y + 3\right) = 13$

Let us isolate $x$ in equation $\left[2\right]$

$\left[2\right] x + 4 \left(y + 3\right) = 13$
$\left[2\right] \implies x = 13 - 4 \left(y + 3\right)$
$\left[2\right] \implies x = 13 - 4 y - 12$
$\left[2\right] \implies x = - 4 y + 1$

We then substitute $x$ in equation $\left[1\right]$ with its equivalent we obtained in equation $\left[2\right]$.

$\left[1\right] 3 \left(x + 5\right) - 2 \left(y - 1\right) = 6$

$\left[1\right] \implies 3 \left(\left(- 4 y + 1\right) + 5\right) - 2 \left(y - 1\right) = 6$
$\left[1\right] \implies 3 \left(- 4 y + 6\right) - 2 \left(y - 1\right) = 6$
$\left[1\right] \implies - 12 y + 18 - 2 y + 2 = 6$
$\left[1\right] \implies - 14 y = 6 - 18 - 2$
$\left[1\right] \implies - 14 y = - 14$
$\left[1\right] \implies y = 1$

To get $x$, replace $y$ with its value in any of the equations above

$x = - 4 y + 1$
$x = - 4 \left(1\right) + 1$

$x = - 4 + 1$

$x = - 3$