-3x + 3y - z = -11  x - 2y - 2z = 3  -2x + 2y + 3z = 0

We are given a system of three equations:

1. \(-3x + 3y - z = -11\)
2. \(x - 2y - 2z = 3\)
3. \(-2x + 2y + 3z = 0\)

Let's solve this system step by step using the substitution or elimination method.

### Step 1: Simplify the equations (if needed).
We can start by simplifying the equations if necessary, but these are already simplified.

### Step 2: Solve one of the equations for one variable.
We can start by solving Equation 2 for \(x\):
\[
x = 2y + 2z + 3
\]

### Step 3: Substitute this expression into the other two equations.

Substitute \(x = 2y + 2z + 3\) into Equations 1 and 3.

#### Substituting into Equation 1:
\[
-3(2y + 2z + 3) + 3y - z = -11
\]
Simplify:
\[
-6y - 6z - 9 + 3y - z = -11
\]
\[
-3y - 7z - 9 = -11
\]
\[
-3y - 7z = -2
\]
\[
3y + 7z = 2 \quad \text{(Equation 4)}
\]

#### Substituting into Equation 3:
\[
-2(2y + 2z + 3) + 2y + 3z = 0
\]
Simplify:
\[
-4y - 4z - 6 + 2y + 3z = 0
\]
\[
-2y - z - 6 = 0
\]
\[
2y + z = -6 \quad \text{(Equation 5)}
\]

### Step 4: Solve the system of two equations.

We now have the system:
1. \(3y + 7z = 2\)
2. \(2y + z = -6\)

Solve Equation 2 for \(z\):
\[
z = -6 - 2y
\]

Substitute this into Equation 1:
\[
3y + 7(-6 - 2y) = 2
\]
Simplify:
\[
3y - 42 - 14y = 2
\]
\[
-11y - 42 = 2
\]
\[
-11y = 44
\]
\[
y = -4
\]

### Step 5: Substitute \(y = -4\) into the expression for \(z\):
\[
z = -6 - 2(-4)
\]
\[
z = -6 + 8 = 2
\]

### Step 6: Substitute \(y = -4\) and \(z = 2\) into the expression for \(x\):
\[
x = 2(-4) + 2(2) + 3
\]
\[
x = -8 + 4 + 3 = -1
\]

### Final Solution:
The solution to the system is:
\[
x = -1, \quad y = -4, \quad z = 2
\]

Would you like further details on any of the steps?

### Related Questions:
1. How do you solve a system of equations using substitution?
2. What are the differences between substitution and elimination methods?
3. Can all systems of linear equations be solved using substitution?
4. What happens if a system of equations has no solutions or infinitely many solutions?
5. How do you solve a system of equations using matrices?

### Tip:
When solving systems of equations, it's helpful to choose the method (substitution, elimination, or matrix methods) that simplifies the calculations early on.

-3x + 3y - z = -11, x - 2y - 2z = 3, -2x + 2y + 3z = 0

Learn how to solve the system of three linear equations -3x + 3y - z = -11, x - 2y - 2z = 3, and -2x + 2y + 3z = 0 using the substitution method. Follow a detailed step-by-step solution that explores key linear algebra concepts. Ideal for students in Grades 9-12.

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