How to do solve system of equations with three variables?

1 Answer
Apr 3, 2015

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