# How do you solve the following system:  1/2x - 2y = 4 , 4x + y = 2 ?

May 18, 2018

See a solution process below:

#### Explanation:

Step 1) Solve the first equation for $x$:

$\frac{1}{2} x - 2 y = 4$

$\textcolor{red}{2} \left(\frac{1}{2} x - 2 y\right) = \textcolor{red}{2} \cdot 4$

$\left(\textcolor{red}{2} \cdot \frac{1}{2} x\right) - \left(\textcolor{red}{2} \cdot 2 y\right) = 8$

$\frac{\textcolor{red}{2}}{2} x - 4 y = 8$

$1 x - 4 y = 8$

$x - 4 y + \textcolor{red}{4 y} = 8 + \textcolor{red}{4 y}$

$x - 0 = 8 + 4 y$

$x = 8 + 4 y$

Step 2) Substitute $\left(8 + 4 y\right)$ for $x$ in the second equation and solve for $y$:

$4 x + y = 2$ becomes:

$4 \left(8 + 4 y\right) + y = 2$

$\left(4 \cdot 8\right) + \left(4 \cdot 4 y\right) + y = 2$

$32 + 16 y + y = 2$

$32 + 16 y + 1 y = 2$

$32 + \left(16 + 1\right) y = 2$

$32 + 17 y = 2$

$32 - \textcolor{red}{32} + 17 y = 2 - \textcolor{red}{32}$

$0 + 17 y = - 30$

$17 y = - 30$

$\frac{17 y}{\textcolor{red}{17}} = - \frac{30}{\textcolor{red}{17}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{17}}} y}{\cancel{\textcolor{red}{17}}} = - \frac{30}{17}$

$y = - \frac{30}{17}$

Step 3) Substitute $- \frac{30}{17}$ for $y$ in the solution to the first equation at the end of Step 1 and calculate $x$:

$x = 8 + 4 y$ becomes:

$x = 8 + \left(4 \cdot - \frac{30}{17}\right)$

$x = 8 + \left(- \frac{120}{17}\right)$

$x = 8 - \frac{120}{17}$

$x = \left(\frac{17}{17} \cdot 8\right) - \frac{120}{17}$

$x = \frac{136}{17} - \frac{120}{17}$

$x = \frac{16}{17}$

The Solution Is:

$x = \frac{16}{17}$ and $y = - \frac{30}{17}$

Or

$\left(\frac{16}{17} , - \frac{30}{17}\right)$