# How do you solve the following system: 7y − 6x − 5 = 0, -5x − y = 14 ?

May 30, 2018

See a solution process below:

#### Explanation:

Step 1) Solve the second equation for $y$:

$- 5 x - y = 14$

$- 5 x - \textcolor{b l u e}{14} - y + \textcolor{red}{y} = 14 - \textcolor{b l u e}{14} + \textcolor{red}{y}$

$- 5 x - 14 - 0 = 0 + y$

$- 5 x - 14 = y$

$y = - 5 x - 14$

Step 2) Substitute $\left(- 5 x - 14\right)$ for $y$ in the first equation and solve for $x$:

$7 y - 6 x - 5 = 0$ becomes:

$7 \left(- 5 x - 14\right) - 6 x - 5 = 0$

$- \left(7 \times 5 x\right) - \left(7 \times 14\right) - 6 x - 5 = 0$

$- 35 x - 98 - 6 x - 5 = 0$

$- 35 x - 6 x - 98 - 5 = 0$

$\left(- 35 - 6\right) x - 103 = 0$

$- 41 x - 103 = 0$

$- 41 x - 103 + \textcolor{red}{103} = 0 + \textcolor{red}{103}$

$- 41 x - 0 = 103$

$- 41 x = 103$

$\frac{- 41 x}{\textcolor{red}{- 41}} = \frac{103}{\textcolor{red}{- 41}}$

$\frac{- \textcolor{red}{\cancel{\textcolor{b l a c k}{41}}} x}{\cancel{\textcolor{red}{- 41}}} = - \frac{103}{41}$

$x = - \frac{103}{41}$

Step 3) Substitute $- \frac{103}{41}$ for $x$ in the solution to the second equation at the end of Step 1 and calculate $y$:

$y = - 5 x - 14$ becomes:

$y = \left(- 5 \times - \frac{103}{41}\right) - 14$

$y = \frac{515}{41} - \left(\frac{41}{41} \times 14\right)$

$y = \frac{515}{41} - \frac{574}{41}$

$y = - \frac{59}{41}$

The Solution Is:

$x = - \frac{103}{41}$ and $y = - \frac{59}{41}$

Or

$\left(- \frac{103}{41} , - \frac{59}{41}\right)$