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#