# How do you solve x = y - 8 and -x - y = 0 using substitution?

Feb 20, 2017

x= -4 , y= 4

#### Explanation:

From the second equation -x-y=0, it is y=-x. Now substitute this in the first equation ,

x= -x-8 $\to$ 2x =-8 $\to$ x= -4

Hence y=4

Feb 20, 2017

$\left(- 4 , 4\right)$

#### Explanation:

$\text{Given } \textcolor{red}{x = y - 8}$ we can $\textcolor{b l u e}{\text{substitute}}$ this directly into the other equation, and solve for y

$\Rightarrow - \left(\textcolor{red}{y - 8}\right) - y = 0$

distributing gives.

$- y + 8 - y = 0$

simplifying.

$- 2 y + 8 = 0$

subtract 8 from both sides of the equation.

$- 2 y \cancel{+ 8} \cancel{- 8} = 0 - 8$

$\Rightarrow - 2 y = - 8$

To solve for y, divide both sides by - 2

$\frac{\cancel{- 2} y}{\cancel{- 2}} = \frac{- 8}{- 2}$

$\Rightarrow y = 4$

To find x, substitute y = 4 into $x = y - 8$

$y = 4 \to x = 4 - 8 = - 4$

$\Rightarrow \left(- 4 , 4\right) \text{ is the solution}$

Feb 20, 2017

Replace $x$ with $\left(y - 8\right)$ in $\text{–} x - y = 0$; solve for $y$.
Use this $y$-value in either equation to solve for $x$.

$\left(x , y\right) = \left(\text{–} 4 , 4\right)$.

#### Explanation:

Each of these equations represents a line in 2D-space. Solving the system of these two equations means finding all the $\left(x , y\right)$ points where the lines cross.

We are given the equations $x = y - 8$ and $\text{–} x - y = 0$. From the first equation, we have a value for $x$ in terms of $y$. That is, for the first line, $x$ is always $y - 8$. To see if this line has any points in common with the other line $\text{–} x - y = 0$, we want to check if an $x$ value of $y - 8$ works for any point on that other line.

So, we substitute $x$ out for $y - 8$ in the second equation:

$\text{ ""–"x" } - y = 0$
$\text{–} \left(y - 8\right) - y = 0$
$\text{ ""–"y+8" } - y = 0$
$\text{ –"2y="–} 8$
$\text{ } y = 4$

So yes—there is a point on the second line where $x$ will be $y - 8$, and that point occurs when $y = 4$.

The only thing left to do is to find the $x$-value for this point. To do that, we can plug $y = 4$ into either of our line equations and solve for $x$. (Since $y = 4$ is where the lines cross, both equations will have the same $x$-value for that $y$).

Using the first equation, we get:

$x = y - 8$
$x = 4 - 8$
$\textcolor{w h i t e}{x} = \text{–} 4$

(Or, using the second equation, we get

$\text{–} x - y = 0$
$\text{–} x - 4 = 0$
$\text{–"x" } = 4$
$\text{ "x="–} 4$

which gives the same $x$-value, as we'd expect.)

So our solution for the system is $\left(x , y\right) = \left(\text{–} 4 , 4\right)$.

graph{(x-y+8)(x+y)=0 [-12.17, 7.83, -2.76, 7.24]}