how to solve 2x+3y+2z=4,3x+4y+5z=-5,3x+6y+3z=-12

Question

i know to solve the linear equation of two variables but three variables are so difficult to me please help me

Answer 1

Label the equations:

(1) 2x+3y+2z=4

(2) 3x+4y+5z=-5

3x+6y+3z=-12⇒(3) x+2y+z=-4, after dividing through by 3.

2×(3)-(1): y=-8-4=-12, so y=-12 and we can create new equations by substituting y=-12 in the equations:

2x-36+2z=4⇒2x+2z=40⇒(4) x+z=20, after dividing through by 2.

(2) becomes 3x-48+5z=-5⇒(5) 3x+5z=43.

We don’t need any more equations because we now have two equations and two unknowns.

3×(4)-(5): -2z=60-43=17, so z=-17/2 and x=20-z=20+17/2=57/2.

SOLUTION: x=57/2, y=-12, z=-17/2.