Three equations

x-y+z=-1
x+y+3z=-3
2x-y+2z=0

x-y+z=-1
x+y+3·z=-3
2·x-y+2·z=0

x-y+z = -1
x+y+3z = -3
2x-y+2z = 0

Pivot: Row 1 ↔ Row 3
2x-y+2z = 0
x+y+3z = -3
x-y+z = -1

Row 2 - 1/2 · Row 1 → Row 2
2x-y+2z = 0
1.5y+2z = -3
x-y+z = -1

Row 3 - 1/2 · Row 1 → Row 3
2x-y+2z = 0
1.5y+2z = -3
-0.5y = -1

Row 3 - -0.5/1.5 · Row 2 → Row 3
2x-y+2z = 0
1.5y+2z = -3
0.667z = -2


z = -2/0.66666667 = -3
y = -3-2z/1.5 = -3-2 · (-3)/1.5 = 2
x = 0+y-2z/2 = 0+2-2 · (-3)/2 = 4

x = 4
y = 2
z = -3

Our linear equations calculator calculates it.