# How do you solve by substitution x + 3y = 5 and 4x + 5y = 13?

Jun 8, 2015

$x + 3 y = 5$
color(blue)(x=5-3y ( we substract $3 y$ on each side )

Now that we have $\textcolor{b l u e}{x}$, we can substitute it in the second equation :

$4 \textcolor{b l u e}{x} + 5 y = 13$
$4 \cdot \textcolor{b l u e}{\left(5 - 3 y\right)} + 5 y = 13$
$20 - 12 y + 5 y = 13$
$20 - 7 y = 13$
$20 = 13 + 7 y$ ( we add $7 y$ on each side )
$20 - 13 = 7 y$
$7 y = 7$
color(red)(y=1 ( we divide by $7$ on each side )

Now that we have $\textcolor{red}{y}$, we can find $\textcolor{b l u e}{x}$ :

$\textcolor{b l u e}{x} = 5 - 3 \textcolor{red}{y}$
$\textcolor{b l u e}{x} = 5 - 3 \cdot 1$
$\textcolor{b l u e}{x} = 5 - 3$
$\textcolor{b l u e}{x = 2}$