# What is the solution to the following system of linear equations: 4x-y=-6 x-2y=-5?

Sep 11, 2015

$\left\{\begin{matrix}x = - 1 \\ y = 2\end{matrix}\right.$

#### Explanation:

Your starting system of equations looks like this

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

Multiply the first equation by $\left(- 2\right)$ to get

$\left\{\begin{matrix}4 x - y = - 6 | \cdot \left(- 2\right) \\ x - 2 y = - 5\end{matrix}\right.$

$\left\{\begin{matrix}- 8 x + 2 y = 12 \\ \text{ } x - 2 y = - 5\end{matrix}\right.$

Notice that if you add the two equations by adding the left-hand sides and the right-hand sides separately, you can eliminate the $y$-term.

The resulting equation will have only one unknown, $x$.

$\left\{\begin{matrix}- 8 x + 2 y = 12 \\ \text{ } x - 2 y = - 5\end{matrix}\right.$
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$- 8 x + \textcolor{red}{\cancel{\textcolor{b l a c k}{2 y}}} + x - \textcolor{red}{\cancel{\textcolor{b l a c k}{2 y}}} = 12 + \left(- 5\right)$

$- 7 x = 7 \implies x = \frac{7}{\left(- 7\right)} = \textcolor{g r e e n}{- 1}$

Plug this value of $x$ into one of the two original equations to get the value of $y$

$4 \cdot \left(- 1\right) - y = - 6$

$- 4 - y = - 6$

$- y = - 2 \implies y = \frac{\left(- 2\right)}{\left(- 1\right)} = \textcolor{g r e e n}{2}$

The solution set for this system of equations will thus be

$\left\{\begin{matrix}x = - 1 \\ y = 2\end{matrix}\right.$