Solve the following system of equations for all three variables.minus, 4, x, minus, 7, y, plus, 4, z, equals, 10−4x−7y+4z=10x, plus, 4, y, minus, 4, z, equals, 8x+4y−4z=87, x, plus, y, minus, z, equals, 27x+y−z=2

The system of equations is:

-4x - 7y + 4z = 10 x + 4y - 4z = 8 7x + y - z = 2

Step 1: Add the first and second equations:

-4x - 7y + 4z + x + 4y - 4z = 10 + 8 -3x - 3y = 18 Divide by -3: x + y = -6

Step 2: Subtract the third equation from the new equation we just found:

x + y - (7x + y - z) = -6 - 2 -6x + z = -8

Step 3: Now we have two new equations:

x + y = -6 -6x + z = -8

We can solve this system by substitution or elimination. Let's use substitution. Solve the first equation for x:

x = -6 - y

Substitute this into the second equation:

-6(-6 - y) + z = -8 36 + 6y + z = -8 6y + z = -8 - 36 6y + z = -44 Divide by 6: y + z/6 = -44/6 y + z/6 = -22/3

Step 4: Now we have two new equations:

x = -6 - y y + z/6 = -22/3

Solve the first equation for y:

y = -6 - x

Substitute this into the second equation:

-6 - x + z/6 = -22/3 -6 - x + z = -44 -x + z = -44 + 6 -x + z = -38 x - z = 38

Step 5: Now we have two new equations:

x = -6 - y x - z = 38

Solve the first equation for z:

z = x - 38

Substitute this into the second equation:

x = -6 - (x - 38) x = -6 - x + 38 2x = 32 x = 16

Substitute x = 16 into the first equation:

16 = -6 - y y = -22

Substitute x = 16 and y = -22 into the third equation:

7(16) + (-22) - z = 2 112 - 22 - z = 2 90 - z = 2 z = 88

So the solution to the system of equations is x = 16, y = -22, z = 88.