# How do you solve 0x – 8y = -16 and -8x + 2y = 36 ?

Jan 20, 2018

$x = - 4$
$y = 2$

#### Explanation:

We can use the elimination method to solve this system.

Equation 1: $0 x - 8 y = - 16$
Equation 2: $- 8 x + 2 y = 36$

In the elimination method, we multiply each equation by a suitable number so that the two equations have a like term. For this problem, we will be focusing on $y$.

First, we will multiply equation 1 by the coefficient of $y$ in equation 2, and multiply equation 2 by the coefficient of $y$ in equation 1.

This means will will be multiplying equation 1 by $\textcolor{red}{2}$ and multiplying equation 2 by $\textcolor{b l u e}{- 8}$.

Equation 1: $\textcolor{red}{2} \setminus \times \left(0 x \textcolor{b l u e}{- 8} y = - 16\right)$
Equation 2: $\textcolor{b l u e}{- 8} \setminus \times \left(- 8 x + \textcolor{red}{2} y = 36\right)$

New Equation 1: $0 x - 16 y = - 32$
New Equation 2: $64 x - 16 y = - 288$

Now that both equations have a like term ($- 16 y$), we can subtract the second equation from the first equation to eliminate them.

$\implies \left(0 x - 16 y = - 32\right) - \left(64 x - 16 y = - 288\right)$

$\implies - 64 x + 0 y = 256$

$\implies - 64 x = 256$

Now all we have to do is divide both sides by -64 to isolate and solve for $x$:

$\frac{- 64 x}{-} 64 = \frac{256}{-} 64$

$\textcolor{m a \ge n t a}{x = - 4}$

We can then substitute $x$ into either equation 1 or equation 2 to solve for $y$. I will be using equation 2.

Substitute $x$ with $- 4$:
$64 \left(- 4\right) - 16 y = - 288$

$- 256 - 16 y = - 288$

Add $256$ to both sides:
$- 256 \textcolor{red}{+ 256} - 16 y = - 288 \textcolor{red}{+ 256}$
$- 16 y = - 32$

Divide both sides by $- 16$ to isolate $y$.
$\frac{- 16 y}{-} 16 = \frac{- 32}{-} 16$

$\textcolor{m a \ge n t a}{y = 2}$

- - Alternate method: - -

As you may have noticed, equation 1 has a term $0 x$. $\textcolor{red}{0 x}$ will equate to $\textcolor{red}{0}$ no matter the value of $x$, so the equation can be converted into:

$- 8 y = - 16$

We can then divide both sides by $- 8$ to solve for $y$:

$\frac{- 8 y}{-} 8 = \frac{- 16}{-} 8$

$\textcolor{m a \ge n t a}{y = 2}$

Then substitute the value of $y$ in to the second equation and solve for $x$:

$- 8 x + 2 \left(2\right) = 36$

$- 8 x + 4 = 36$

Subtract 4 from both sides of the equation:

$- 8 x + 4 \textcolor{red}{- 4} = 36 \textcolor{red}{- 4}$

$- 8 x = 32$

Then divide both sides by -8 to solve for $x$:

$\frac{- 8 x}{-} 8 = \frac{32}{-} 8$

$\textcolor{m a \ge n t a}{x = - 4}$

The point $\left(- 4 , 2\right)$ is the point of intersection between the two lines.

graph{(-8y+16)(2y-8x-36)=0 [-12.66, 12.65, -6.33, 6.33]}

Feb 6, 2018

$0 x$ simply means $0$... so you can add as many variables as you want if there's a $0$ behind them...

So... in the first equation
$\implies 0 x - 8 y = - 16$
$\implies 0 - 8 y = - 16$
$\implies - 8 y = - 16$
Cancel out the minuses
$\implies 8 y = 16$
$\implies y = \frac{16}{8}$
=>color(red)(y=2
Put this value in the second equation

Second equation
$- 8 x + 2 y = 36$
Put value
$- 8 x + 2 \times 2 = 36$
Multiply
$- 8 x + 4 = 36$
Transfer the value 4
$- 8 x = 32$
Transfer $- 8$
$x = \frac{32}{- 8}$
color(red)(x=-4