Question

How do you solve the system of equations `3x + 6y = - 18` and `- 6x + y = - 16`?

Answer 1

`(2,-4)`

Explanation:

There are two main methods used to solve systems of equations without graphs or matrices:

1. Solve for one of the variables, and then plug in and solve and plug back in and solve for the second variable, or

2. Multiply one or both of the equations until one of the variables in both is equal or opposite.

Method 1:

In this case, we already almost have `y` isolated in the second equation, so we'll finish that:

`-6x + y = -16`

Add 6x to both sides:

`y=-16+6x`

Since `y` and `x` are equal in both equations, you can replace any `y` in the first equation with what we determined `y` to be equal to, in terms of `x`:

`3x+6y=-18 -> 3x+6(-16+6x)=-18`

You can distribute the 6, which simplifies to:

`3x-96+36x=-18`

Which simplifies to:

`39x-96=-18`

Now, we can solve for `x`:

`39x-96=-18`
`39x=78`
`x=2`

With `x`, we can simply plug it into either equation and then solve for `y`:

`3(2)+6y=-18`
`6+6y=-18`
`6y=-24`
`y=-4`

Now if you plug in `2` for `x` and `-4` for `y` in either equation, both should turn out true:

`3(2)+6(-4)=-18`
`6+-24=-18`
`-18=-18`

It works for one! We just need to plug it into the other equation to be 100% sure:

`-6(2)+(-4)=-16`
`-12+-4=-16`
`-16=-16`

Method 2:

Canceling out variables works equally well. But in order to do that, we need to
1. Decide which variable to cancel first, and
2. Multiply one or both equations to get them to be equal or opposite.

In this situation, `x` is the easiest to cancel, because you only need to multiply one equation by a factor of `2`, which keeps the rest more simple.

We start with:
`3x+6y=-18`
`-6x+y=-16`

We want the `x`'s to be equal, so we'll go ahead an multiply all terms in the top equation by 2:

`6x+12y=-36`
`-6x+y=-16`

This might feel weird, but verically add these two equations to get:

`0x+13y=-52`

Which simplifies to:

`13y=-52`
`y=-4`

Now that we have `y`, let's plug it in and solve, then check both equations:

`-6x+(-4)=-16`
`-6x-4=-16`
`-6x=-12`
`x=2`

Excellent! Let's plug in both the `x` and `y` values to check we did it right:

`3(2)+6(-4)=-18`
`6+-24=-18`
`-18=-18`

It works for one! We just need to plug it into the other equation to be 100% sure:

`-6(2)+(-4)=-16`
`-12+-4=-16`
`-16=-16`