-x1 + 2x2 - 2x3 = -3; x1 - x2 + x3 = 2; x1 + x2 - 3x3 = -6

Steps to Solve

1. Write down the equations
   The given system of equations is:
   $$
   -x_1 + 2x_2 - 2x_3 = -3 \quad (1)
   $$
   $$
   x_1 - x_2 + x_3 = 2 \quad (2)
   $$
   $$
   x_1 + x_2 - 3x_3 = -6 \quad (3)
   $$

2. Isolate one variable
   Start with equation (2) to isolate $x_1$:
   $$
   x_1 = x_2 - x_3 + 2 \quad (4)
   $$

3. Substitute into other equations
   Substitute equation (4) into equations (1) and (3):

For equation (1):
$$
-(x_2 - x_3 + 2) + 2x_2 - 2x_3 = -3
$$
Simplifying gives:
$$
-x_2 + x_3 - 2 + 2x_2 - 2x_3 = -3
$$
Combine like terms:
$$
x_2 - x_3 - 2 = -3
$$
This simplifies to:
$$
x_2 - x_3 = -1 \quad (5)
$$

For equation (3):
$$
(x_2 - x_3 + 2) + x_2 - 3x_3 = -6
$$
Simplifying gives:
$$
x_2 - x_3 + 2 + x_2 - 3x_3 = -6
$$
This simplifies to:
$$
2x_2 - 4x_3 + 2 = -6
$$
Removing 2 from both sides:
$$
2x_2 - 4x_3 = -8
$$
Dividing by 2 gives:
$$
x_2 - 2x_3 = -4 \quad (6)
$$

4. Solve the new equations
   You now have two equations, (5) and (6):
   From (5):
   $$
   x_2 = x_3 - 1 \quad (7)
   $$

Substitute (7) into (6):
$$
(x_3 - 1) - 2x_3 = -4
$$
This simplifies to:
$$
-x_3 - 1 = -4
$$
So:
$$
-x_3 = -3 \quad \Rightarrow x_3 = 3
$$

5. Find remaining variables
   Substituting $x_3 = 3$ back into (7):
   $$
   x_2 = 3 - 1 = 2
   $$

Finally, substitute $x_2$ and $x_3$ back into (4) to find $x_1$:
$$
x_1 = 2 - 3 + 2 = 1
$$

6. Final solution
   The solution is:
   $$
   x_1 = 1, \quad x_2 = 2, \quad x_3 = 3
   $$