Question 123854
There are lot of ways to solve equations with 3 unknown variable. You can do matrices or substitution. For the sake of easiness, I will demonstrate how to solve the 3 equations using substitution.

Equations:
{{{2x+y-3z=-12}}}
{{{5x-y+z=11}}}
{{{x+3y-2z=-13}}}

Isolate the y in the first variable.
{{{y=-2x+3z-12}}}
Plug the y in the second equation.
{{{5x-(-2x+3z-12)+z=11}}}
Simplify.
{{{7x-2z+12=11}}}
Isolate z.
{{{z=(7/2)x+(1/2)}}}
Plug the z in the y.
{{{y=-2x+3((7/5)x+(1/2))-12}}}
Simplify.
{{{y=8.5x-10.5}}}
Plug the NEW y and z value in the last equation.
{{{x+3(8.5x-10.5)-2((7/2)x+(1/2))=-13}}}
Simplify and solve for x.
{{{x=1}}}
Plug x=1 to the new y.
{{{y=8.5(1)-10.5}}}
{{{y=-2}}}
Plug the y and x value to any of the equations. you should get {{{z=4}}}
Solutions:
{{{x=1}}}
{{{y=-2}}}
{{{z=4}}}