Question

How to do solve system of equations with three variables?

Answer 1

If there are `3` variables, then there must be `3` equations.

Lets say `A, B, C` are our equations and `x, y, z` are the variables.

You will follow these three steps:

• By using `C`, write `z` in terms of `x` and `y`

• Replace `z` with its equivalent in `B`. Then write `y` in terms of `x`

• In `A`, replace `y` with its equivalent and replace `z` with its equivalent (if its equivalent involves `y`, replace `y`) then solve `A` for `x`.

Now you should know the value of `x`. You should have written `y` in terms of `x` so plug `x` and you will find `y`.

Finally, you should have written `z` in terms of `x` and `y` so you can find the value of `z`.

Example

`A: x+y+z=10`

`B: 2x+y+z=12`

`C: 3x+2y+z=17`

Lets find `x, y, z`

We are writing `z` in terms of `x` and `y` by using `C`, and I will call this equation as `1'`

`z=17-3x-2y`

Now we are plugging `1'` to `B`

`2x+y+(17-3x-2y)=12`
`-x-y=-5`

So we can write `y` in terms of `x`. I will call this equation as `2'`

`y=5-x`

Now we are plugging `1'` and `2'` to `A`. (We also replaced `y` in `1'` by using `2'`)

`x+(5-x) +(17-3x-2(5-x))=10`

`5+17-3x-10+2x=10`

`-x=-2->x=2`

Now we know the value of `x`. So:

By using `2'`, `y=3`

By using `1'`, `z=17-3*2-2*(3) = 5`

So `x=2, y=3, z=5`