# How do you solve the system of equations x- y = 7 and - 2x + 5y = - 8?

Aug 7, 2017

By manipulating the first equation and combining it with the second, we can eventually arrive at the answer: x = 9, y = 2.

#### Explanation:

Your goal here is to remove one of the variables from the problem. You can see that the first equation has x and the second equation has -2x. If we double the first equation, we get:

$2 x - 2 y = 14$

Then we simply add that to the second equation:
$2 x - 2 y = 14$
+$- 2 x + 5 y = - 8$

$3 y = 6$

The positive 2x and the negative 2x cancel out, leaving us with just the 3y = 6.

Divide both sides by 3 and we get y=2.

Last, just plug 2 in for y in either equation (I'll choose the first since it's simpler):

$x - 2 = 7$

Add 2 to both sides to get x=9.

So our final answer is x = 9, y = 2.

Aug 7, 2017

You will need to solve one of the variables using the substitution method.

#### Explanation:

Begin by solving either the $x$ or $y$ variable by first eliminating or canceling out one of the variables. We can then plug in that variable into the first equation and solve for the second equation

$x - y = 7 \text{ " and " } - 2 x + 5 y = - 8$

To Solve for $y$ in the second equation, start by multiplying the first equation by $2$ and add the result to second in order to cancel out the $x$ variable

$2 \cdot \left(x - y = 7\right) = 2 x - 2 y = 14 \to$ Add this to the second equation

$- 2 x + 5 y = - 8$
$+ 2 x - 2 y = 14$
$\ldots \ldots \ldots \ldots \ldots \ldots \ldots .$

$3 y = 6$

$y = 2$

Now plug in the $2$ for $y$ in the first equation to solve for $x$

$x - 2 + 2 = 7 + 2$

$x = 9$

We have now solved both variables. Check to make sure both equations are equal.

Aug 7, 2017

The point of intersection between the two lines is $\left(9 , 2\right)$.

Refer to the explanation for the process.

#### Explanation:

Solve system of equations:

These are linear equations. Since they are a system, both equations are solved simultaneously by substitution. The resulting values for x and y is the point at which the two lines intersect on a graph.

The two equations are:

$x - y = 7$$\textcolor{w h i t e}{\ldots} \mathmr{and} \textcolor{w h i t e}{\ldots}$$- 2 x + 5 y = - 8$

First Equation: $x - y = 7$

$x - y = 7$

Add $y$ to both sides.

$x = 7 + y$

Second Equation: $- 2 x + 5 y = - 8$

Substitute $7 + y$ for $x$ in the equation.

$- 2 \left(7 + y\right) + 5 y = - 8$

Simplify.

$- 14 - 2 y + 5 y = - 8$

Add $14$ to both sides.

$- 2 y + 5 y = - 8 + 14$

Simplify.

$3 y = 6$

Divide both sides by $3$.

$y = 2$

Now substitute the value of $y$ back into the first equation and solve for $x$.

$x - 2 = 7$

Add $2$ to both sides.

$x = 9$

The point of intersection is $\left(9 , 2\right)$.