Question

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

Answer 1

`x = -4`
`y = 2`

Explanation:

We can use the elimination method to solve this system.

Equation 1: `0x - 8y = -16`
Equation 2: `-8x + 2y = 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 `color(red)2` and multiplying equation 2 by `color(blue)(-8)`.

Equation 1: `color(red)2 \times (0x color(blue)( - 8)y = -16)`
Equation 2: `color(blue)(-8) \times (-8x + color(red)2y = 36)`

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

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

`=> (0x - 16y = -32) - (64x -16y = -288)`

`=> -64x +0y = 256`

`=> -64x = 256`

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

`(-64x)/-64 = (256)/-64`

`color(magenta)(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(-4) -16y = -288`

`-256 -16y = -288`

Add `256` to both sides:
`-256 color(red)(+ 256) -16y = -288color(red)(+ 256)`
`-16y = -32`

Divide both sides by `-16` to isolate `y`.
`(-16y)/-16 = (-32)/-16`

`color(magenta)(y = 2)`

- - Alternate method: - -

As you may have noticed, equation 1 has a term `0x`. `color(red)(0x)` will equate to `color(red)0` no matter the value of `x`, so the equation can be converted into:

`-8y = -16`

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

`(-8y)/-8 = (-16)/-8`

`color(magenta)(y = 2)`

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

`-8x + 2(2) = 36`

`-8x + 4 = 36`

Subtract 4 from both sides of the equation:

`-8x + 4 color(red)(-4)= 36 color(red)(-4)`

`-8x = 32`

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

`(-8x)/-8 = (32)/-8`

`color(magenta)(x = -4)`

The point `(-4,2)` is the point of intersection between the two lines.

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

Answer 2

`0x` 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
`=>0x-8y=-16`
`=>0-8y=-16`
`=>-8y=-16`
Cancel out the minuses
`=>8y=16`
`=>y=16/8`
`=>color(red)(y=2`
Put this value in the second equation

Second equation
`-8x+2y=36`
Put value
`-8x+2xx2=36`
Multiply
`-8x+4=36`
Transfer the value 4
`-8x=32`
Transfer `-8`
`x=32/(-8)`
`color(red)(x=-4`