Question

Does the following system of equations have a solution? Of so, find one. If not, explain why.

2x + y+ z=4
x- y+ 3z=-2
-x + y + z= -2

Answer 1

The given system of equations is:
2x + y + z = 4
x - y + 3z = -2
-x + y + z = -2

We can add the second and third equations to eliminate y and obtain:
0x + 0y + 4z = -4
Simplifying, we get:
z = -1

Substituting z = -1 into the third equation, we obtain:
-x + y - 1 = -2
Simplifying, we get:
x - y = 1

Substituting z = -1 into the first equation, we obtain:
2x + y - 1 = 4
Simplifying, we get:
2x + y = 5

We can add the equations x - y = 1 and 2x + y = 5 to eliminate y and obtain:
3x = 6
Simplifying, we get:
x = 2

Substituting x = 2 into the equation x - y = 1, we obtain:
2 - y = 1
Simplifying, we get:
y = 1

Therefore, the system of equations has a unique solution of (x, y, z) = (2, 1, -1).