Question

How do you solve this system of equations `y= \frac { - 3} { 4} x , x - 4y = 32`?

Answer 1

`(x,y)=(8,-6)`

Explanation:

Given
[1]`color(white)("XXX")y=(-3)/4x`
[2]`color(white)("XXX")x-4y=32`

Using [1] we can substitute `(-3)/4x` for `y` in [2] to get
[3]`color(white)("XXX")x-4xx(-3)/4x=32`

Simplifying [4]
[5]`color(white)("XXX")x-(-3)x=32`

Continuing the simplification:
[6]`color(white)("XXX")4x=32`

Dividing both sides of [6] by `4`
[7]`color(white)("XXX")x=8`

Using [7] we can substitute `8` for `x` in [1] to get
[8]`color(white)("XXX")y=(-3)/4xx 8`

Simplifying [8]
[9]`color(white)("XXX")y=-6`

Answer 2

`(x,y)to(8,-6)`

Explanation:

`color(red)(y)=-3/4xto(1)`

`x-4color(red)(y)=32to(2)`

`"substitute "y=-3/4x" in "(2)`

`rArrx-(4xx-3/4)=32`

`rArrx+3x=32`

`rArr4x=32`

`"divide both sides by 4"`

`rArrx=8`

`"substitute this value in "(1)`

`y=-3/4xx8=-6`

`color(blue)"As a check"`

`"substitute these values in "(2)`

`8+24=32larr" True"`

`rArr"point of intersection "=(8,-6)` graph{(y+3/4x)(y-1/4x+8)((x-8)^2+(y+6)^2-0.06)=0 [-12.49, 12.48, -6.25, 6.24]}

Answer 3

Substitution.

`x=8`
`y=-6`

Explanation:

There are many ways to solve systems of equations. For this system:
`color(green)y=-3/4x` (Eq. 1)
`x-4y=32` (Eq. 2),
it would be easiest to solve it with substitution since Equation (Eq.) 1 is already solved for `y`. This means we can simply plug in the `y` value in the second equation.

This is Eq 2:

`x-4color(green)y=32`

If we plug in Eq. 1 into Eq. 2, we get:

`x-4(-3/4x)=32`
`x+3x=32`
`4x=32`
`x=8`

Now we solved for the first variable. To solve for `y`, all we do is plug in our value of `x` back into Eq. 1:

`y=-3/4x`
`y=-3/4(8)=-6`

So, the solution to the system of equations is:

`x=8`
`y=-6`

To check this answer, you can plug in the `x` and `y` values into Eq.1 and Eq. 2 respectively to see if the equation solves correctly:

Eq 1 verification by plugging in the `x` and `y` values:
`y=-3/4x`
`-6=-3/4(8)=-6` (so it works)
Eq 2 verification by plugging in the `x` and `y` values:
`x-4y=32`
`8-4(-6)=8+24=32` (so it works)