How do I solve a system of linear equations when there is three variables and three equations?

do you know guassian elimination?

o.k., no problem.

(I know this sounds complicated, but you must have done something similar when you had 2 equations with 2 unknowns. )

(1)solve first equation for x in terms of y & z

plug this expression into 2cd equation where you see an x

(2) solve this equation for y in terms of z

now in (1) you can replace y with what you found in (2)
so you have x only in terms of z

and, from (2) you have y in terms of z

choose any original equation and use your expressions found to substitute for x & y in terms of z and so you will have an equation with only z.

Solve for z

now, you can work backwards using (1)  & (2) to solve for y & x

When you're finished, check the solution by plugging x,y,z,values into all three equations