𝑥 − 𝑦 − 𝑧 = 2
3𝑥 − 3𝑦 + 2𝑧 = 16
2𝑥 − 𝑦 + 𝑧 = 9

We are given a system of three linear equations:

1. \( x - y - z = 2 \)
2. \( 3x - 3y + 2z = 16 \)
3. \( 2x - y + z = 9 \)

### Step 1: Solve for \( z \) from equation (1)
From equation (1):
\[
x - y - z = 2 \quad \Rightarrow \quad z = x - y - 2
\]
Substitute this value of \( z \) into equations (2) and (3).

### Step 2: Substitute \( z = x - y - 2 \) into equation (2)
\[
3x - 3y + 2(x - y - 2) = 16
\]
Simplify:
\[
3x - 3y + 2x - 2y - 4 = 16
\]
\[
5x - 5y - 4 = 16
\]
\[
5x - 5y = 20 \quad \Rightarrow \quad x - y = 4
\]
Thus, we now have:
\[
x = y + 4
\]

### Step 3: Substitute \( x = y + 4 \) into equation (3)
Substitute into equation (3):
\[
2x - y + z = 9
\]
Substituting \( x = y + 4 \) and \( z = x - y - 2 \):
\[
2(y + 4) - y + (y + 4 - y - 2) = 9
\]
Simplify:
\[
2y + 8 - y + 2 = 9
\]
\[
y + 10 = 9
\]
\[
y = -1
\]

### Step 4: Solve for \( x \) and \( z \)
Now that \( y = -1 \), substitute into \( x = y + 4 \):
\[
x = -1 + 4 = 3
\]
Substitute \( x = 3 \) and \( y = -1 \) into \( z = x - y - 2 \):
\[
z = 3 - (-1) - 2 = 3 + 1 - 2 = 2
\]

### Solution:
The solution to the system of equations is:
\[
x = 3, \quad y = -1, \quad z = 2
\]

Would you like further details on the steps or have any questions?

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Here are 5 related questions to consider:
1. How can you verify the solution by substituting into the original equations?
2. What is the geometric interpretation of solving a system of three linear equations in three variables?
3. How would the solution change if one of the equations were dependent on the others?
4. How would you solve the system using matrix methods like Gaussian elimination?
5. What happens if the system has no solutions or infinitely many solutions?

**Tip:** Always check the consistency of your solution by substituting it back into the original equations!

Learn how to solve the system of linear equations x - y - z = 2, 3x - 3y + 2z = 16, and 2x - y + z = 9 using substitution and simplification methods. This detailed step-by-step solution covers important algebraic concepts and is suitable for students in Grades 9-12.

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