# How do you solve 2m-5n=-6 and 2m-7n=-14?

Sep 7, 2015

$\left\{\begin{matrix}n = 4 \\ m = 7\end{matrix}\right.$

#### Explanation:

Your system of equations looks like this

$\left\{\begin{matrix}2 m - 5 n = - 6 \\ 2 m - 7 n = - 14\end{matrix}\right.$

You can solve this by substituting $2 m$ from the first equation into the second equation and solving for $n$, or by multplying the first equation by $\left(- 1\right)$, adding the resulting equation to the second equation, and then solving for $n$.

Here's how that first method would look.

$2 m = - 6 + 5 n$

$\left(- 6 + 5 m\right) - 7 n = - 14$

$5 m - 7 n = - 14 + 6$

$- 2 n = - 8 \implies n = \frac{\left(- 8\right)}{\left(- 2\right)} = 4$

This means that $m$ is equal to

$2 m = - 6 + 5 \cdot \left(4\right)$

$m = \frac{14}{2} = 7$

The solution set will thus be

$\left\{\begin{matrix}n = 4 \\ m = 7\end{matrix}\right.$

Now try the second method.

$\left\{\begin{matrix}2 m - 5 n = - 6 | \left(- 1\right) \\ 2 m - 7 n = - 14\end{matrix}\right.$

$\left\{\begin{matrix}- 2 m + 5 n = 6 \\ 2 m - 7 n = - 14\end{matrix}\right.$
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$- \textcolor{red}{\cancel{\textcolor{b l a c k}{2 m}}} + 5 n + \textcolor{red}{\cancel{\textcolor{b l a c k}{2 m}}} - 7 n = 6 + \left(- 14\right)$

$- 2 n = - 8 \implies n = 4$

Once again, $m = 7$ and $n = 4$.