# How do you solve the system x/2-y/3=5/6 and x/5-y/4=71/10?

Mar 21, 2016

Scale and combine the equations to eliminate $y$ and solve for $x$, then substitute that value of $x$ back in to solve for $y$ and find:

$\left\{\begin{matrix}x = - 37 \\ y = - 58\end{matrix}\right.$

#### Explanation:

Given:

$\left\{\begin{matrix}\frac{x}{2} - \frac{y}{3} = \frac{5}{6} \\ \frac{x}{5} - \frac{y}{4} = \frac{71}{10}\end{matrix}\right.$

Multiply the first equation by $6$ and the second by $20$ to get:

$\left\{\begin{matrix}3 x - 2 y = 5 \\ 4 x - 5 y = 142\end{matrix}\right.$

To eliminate $y$, multiply the first of these by $5$, the second by $2$ and subtract the second from the first...

$\left\{\begin{matrix}15 x - 10 y = 25 \\ 8 x - 10 y = 284\end{matrix}\right.$

$\implies 7 x = 25 - 284 = - 259$

Then divide both ends by $7$ to find:

$x = - \frac{259}{7} = - 37$

Substitute this value of $x$ into the equation $3 x - 2 y = 5$ to get:

$- 111 - 2 y = 5$

Add $111$ to both sides to get:

$- 2 y = 116$

Divide both sides by $- 2$ to get:

$y = - 58$