Question

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

Answer 1

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:

`2x - 2y = 14`

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

`3y=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.

Answer 2

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 " " and " " - 2x + 5y = -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 * ( x - y = 7) = 2x - 2y = 14 ->` Add this to the second equation

`- 2x + 5y = -8`
`+2x - 2y = 14`
`......................`

`3y = 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.

Answer 3

The point of intersection between the two lines is `(9,2)`.

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` `color(white)(...)andcolor(white)(...)` `-2x+5y=-8`

First Equation: `x-y=7`

`x-y=7`

Add `y` to both sides.

`x=7+y`

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

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

`-2(7+y)+5y=-8`

Simplify.

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

Add `14` to both sides.

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

Simplify.

`3y=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 `(9,2)`.