Question

How do you solve the system of equations `-3x - 10y = 6` and `- 3x - 3y = - 15`?

Answer 1

`x=8, y=-3`

Explanation:

Our main goal with systems of equations is to cancel one of the variables, so we can solve for the other. Let's just examine our system of equations:

`-3x-10y=6`
`-3x-3y=-15`

We want to manipulate one of the equations so we can cancel a variable. The easiest may be to multiply the second equation by -1.

`-3x-10y=6`
`-1(-3x-3y=-15)`

Distributing the -1, we get our new system being equal to:

`-3x-10y=6`
`3x+3y=15`

At this point, we can add the two equations to get:

`-7y=21` (Now we can solve for `y`)

`color(blue)(y=-3)`

We know that `y=-3`. Now we can plug this back in for y into either of the equations to solve for `x`. Let's plug it into the first equation:

`-3x-10(-3)=6`

`-3x+30=6`

`-3x=6-30`

`-3x=-24`

`color(purple)(x=8)`

We now have our solutions, `x=8` and `y=-3`. To confirm this, we can plug both values in for either equation. Plugging x and y into the second equation, we get:

`-3(8)-3(-3)= -24+9= -15`

The original second equation was `-3x-3y=-15`, and plugging in `x` and `y`, we got `-15` as an answer.

Answer 2

`x = -11/2`
`y = 21/2`

Explanation:

`-3x - 10y = 6`
`-3x - 3y = -15`

First, let's look at the 2nd equation. Let's get `y` by itself:
`-3x - 3y = -15`

Add `3x` to both sides of the equation
`-3y = -15 + 3x`

Divide both sides by `-3`
`y = 5 - x`

Now we substitute the value we got for y back into the first equation:
`-3x - 10y = 6`

Substitute `(5-x)` for `y`
(don't forget the parenthesis! they are VERY important!!!)
`-3x - (5-x) = 6`

Distribute the negative sign to both `5` and `-x`:
`-3x - 5 + x = 6`

Group like terms:
`-2x = 11`

Divide both sides by `-2`
`x = -11/2`

---

Now that we have the value of `x`, we can find the value of `y` by plugging back `x` into one of the equations. Since we simplified the second equation to `y = 5-x`, I'm going to use that:
`y = 5 - x`
`y = 5 - (-11/2)`
`y = 5 + 11/2`
`y = 5 11/2`
`y = 21/2` or `10 1/2`

If you want to be sure the answer is right, all you have to do is plug in the `x` and `y` values back into one of the original equations to see if they make sense.

---

This is just one way to solve system of equations problems. You can also use elimination or other ways.

If you need more examples or help with these types of problems, feel free to watch this video:

Hope this helps!