# How do you solve 5x+6y=24 and 3x+5y=18?

Sep 6, 2015

$\left\{\begin{matrix}x = \frac{12}{7} \\ y = \frac{18}{7}\end{matrix}\right.$

#### Explanation:

Your starting system of equations is

$\left\{\begin{matrix}5 x + 6 y = 24 \\ 3 x + 5 y = 18\end{matrix}\right.$

Multiply the first equation by $\left(- 3\right)$ and the second equation by $5$ to get

$\left\{\begin{matrix}5 x + 6 y = 24 | \cdot \left(- 3\right) \\ 3 x + 5 y = 18 | \cdot 5\end{matrix}\right.$

$\left\{\begin{matrix}- 15 x - 18 y = - 72 \\ 15 x + 25 y = 90\end{matrix}\right.$

Notice that if you add these two equations, more specifically if you add the left-hand sides and the right-hand sides separately, you can eliminate the $x$-term.

This will allow you to solve the resulting equation for $y$

$\left\{\begin{matrix}- 15 x - 18 y = - 72 \\ 15 x + 25 y = 90\end{matrix}\right.$
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$\textcolor{red}{\cancel{\textcolor{b l a c k}{15 x}}} - 18 y + \textcolor{red}{\cancel{\textcolor{b l a c k}{15 x}}} + 25 y = - 72 + 90$

$7 y = 18 \implies y = \textcolor{g r e e n}{\frac{18}{7}}$

Now use this value of $y$ in one of the two original equations to get the value of $x$

$5 x + 6 \cdot \frac{18}{7} = 24$

$35 x + 108 = 168$

$35 x = 60 \implies x = \textcolor{g r e e n}{\frac{12}{7}}$