Question

4x-3y+z=0

-2y-5z=6

-4x+5y+2z=-2

Answer 1

4x-3y+z=0 -2y-5z=6 -4x+5y+2z=-2

1) 4x - 3y + z = 0
2) -2y - 5z = 6
3) -4x + 5y + 2z = -2

Add equation 3 to equation 1. That will eliminate the x,
giving us an equation with only y and z, which we can
use with equation 2.

    4x -  3y +   z =  0
+(-4x + 5y + 2z = -2)
--------------------------
            2y + 3z =  -2
4) 2y + 3z = -2

Add equation 4 to equation 2, eliminating the y.

  -2y -  5z =  6
+(2y + 3z = -2)
------------------
          -2z =  4
5) -2z = 4
z = -2     <<<<<<<<<<<<<<<<<<<<

Substitute that into equation 2 and solve for y.

-2y - 5z = 6
-2y - 5(-2) = 6
-2y + 10 = 6
-2y = 6 - 10
-2y = -4
y = 2     <<<<<<<<<<<<<<<<<<<<

Substitute both y and z into equation 3 to solve for x.

-4x + 5y + 2z = -2
-4x + 5(2) + 2(-2) = -2
-4x + 10 - 4 = -2
-4x + 6 = -2
-4x = -2 - 6
-4x = -8
x = 2     <<<<<<<<<<<<<<<<<<<<

Verify by substituting all three values into equation 1.

4x - 3y + z = 0
4(2) - 3(2) + (-2) = 0
8 - 6 - 2 = 0
0 = 0

Answer: x = 2, y = 2 and z = -2

Answer 2

From the first equation, z=3y-4x. Substitute this in the other equations:

-2y-5(3y-4x)=6, -17y+20x=6 (1)

-4x+5y+2(3y-4x)=-2, 11y-12x=-2 (2)

(1)×3=-51y+60x=18, (2)×5=55y-60x=-10

Add these together to eliminate x: 4y=8, y=2, 22-12x=-2, 12x=24, x=2, z=3y-4x=-2.

SOLUTION (x,y,z)=(2,2-2)