How do you solve 5x + y = -2 and 4x + 7y = 2?

Sep 8, 2015

$\left\{\begin{matrix}x = - \frac{16}{31} \\ y = \frac{18}{31}\end{matrix}\right.$

Explanation:

Take a look at your system of equations

$\left\{\begin{matrix}5 x + y = - 2 \\ 4 x + 7 y = 2\end{matrix}\right.$

Notice that if you multiply the first equation by $\left(- 7\right)$ and add the left-hands sides and the right-hand sides of the two equations, you can eliminate the $y$-term.

This will leave you with one equation with one unknown, $x$.

$\left\{\begin{matrix}5 x + y = - 2 | \cdot \left(- 7\right) \\ 4 x + 7 y = 2\end{matrix}\right.$

$\left\{\begin{matrix}- 35 x - 7 y = 14 \\ 4 x + 7 y = 2\end{matrix}\right.$
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$- 35 x - \textcolor{red}{\cancel{\textcolor{b l a c k}{7 y}}} + 4 x + \textcolor{red}{\cancel{\textcolor{b l a c k}{7 y}}} = 14 + 2$

$- 31 x = 16 \implies x = \frac{16}{\left(- 31\right)} = \textcolor{g r e e n}{- \frac{16}{31}}$

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

$5 \cdot \frac{- 16}{31} + y = - 2$

$y = - 2 + \frac{80}{31}$

$y = \textcolor{g r e e n}{\frac{18}{31}}$