# How do solve the following linear system?:  4x-2y=2 , x-4y=4 ?

Jun 11, 2017

See a solution process below:

#### Explanation:

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

$x - 4 y = 4$

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

$x - 0 = 4 + 4 y$

$x = 4 + 4 y$

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

$4 x - 2 y = 2$ becomes:

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

$\left(4 \times 4\right) + \left(4 \times 4 y\right) - 2 y = 2$

$16 + 16 y - 2 y = 2$

$16 + \left(16 - 2\right) y = 2$

$16 + 14 y = 2$

$- \textcolor{red}{16} + 16 + 14 y = - \textcolor{red}{16} + 2$

$0 + 14 y = - 14$

$14 y = - 14$

$\frac{14 y}{\textcolor{red}{14}} = - \frac{14}{\textcolor{red}{14}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{14}}} y}{\cancel{\textcolor{red}{14}}} = - 1$

$y = - 1$

Step 3) Substitute $- 1$ for $y$ in the solution to the second equation at the end of Step 1 and calculate $x$:

$x = 4 + 4 y$ becomes:

$x = 4 + \left(4 \cdot - 1\right)$

$x = 4 + \left(- 4\right)$

$x = 4 - 4$

$x = 0$

The solution is: $x = 0$ and $y = - 1$ or $\left(0 , - 1\right)$

Jun 11, 2017

$x = 0$ and $y = - 1$

#### Explanation:

$\textcolor{w h i t e}{-} 4 x - 2 y = 2$
$\textcolor{w h i t e}{-}$
$\textcolor{w h i t e}{-} x - 4 y = 4$

Let's try elimination! If we multiply the first equation by $2$, the $y$s will have equal coefficients, and we'll be able to eliminate them from the set:

$\textcolor{w h i t e}{-} 2 \left(4 x - 2 y = 2\right)$
$\textcolor{w h i t e}{-}$
$\textcolor{w h i t e}{- 2. .} x - 4 y = 4$

color(white(--)

$\textcolor{w h i t e}{-} 8 x - 4 y = 4$
$\textcolor{b l a c k}{-}$
$\textcolor{w h i t e}{-} x - 4 y = 4$
...........................................
$7 x = 0$

divide by $7$ on both sides

$x = 0$

Now let's solve for $y$!

$x - 4 y = 4$

substitute $0$ for $x$

$\left(0\right) - 4 y = 4$

$- 4 y = 4$

divide by $- 4$ on both sides

$y = - 1$

To check our work, let's substitute $x$ and $y$ for $0$ and $- 1$ in the other equation.

$4 x - 2 y = 2$

$4 \left(0\right) - 2 \left(- 1\right)$ should equal $2$, if we did our math right...

$0 - - 2$

$2 = 2$
We were right!!!

$x = 0 , y = - 1$

Just to triple check, let's graph the equations and see where they intercept:

They cross at $\left(0 , - 1\right)$, where $x = 0$ and $y = - 1$