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The system of equations you have provided is:

\[
\begin{aligned}
1) \quad 2x + y - z &= 3 \\
2) \quad x + 3y + 2z &= 14 \\
3) \quad 3x - 2y + z &= -1 \\
\end{aligned}
\]

We need to find \(x + y + z\).

Let's solve this system of equations step by step.

### Step 1: Solve for one variable
I'll first eliminate one variable by adding or subtracting equations.

From equation (1):
\[
2x + y - z = 3 \quad \text{or} \quad z = 2x + y - 3
\]
Substitute this value of \(z\) into equations (2) and (3).

### Step 2: Substitution in equations (2) and (3)
Substitute \(z = 2x + y - 3\) into equation (2):
\[
x + 3y + 2(2x + y - 3) = 14
\]
Simplify:
\[
x + 3y + 4x + 2y - 6 = 14
\]
\[
5x + 5y - 6 = 14
\]
\[
5x + 5y = 20
\]
\[
x + y = 4 \quad \text{(Equation A)}
\]

Now substitute \(z = 2x + y - 3\) into equation (3):
\[
3x - 2y + (2x + y - 3) = -1
\]
Simplify:
\[
3x - 2y + 2x + y - 3 = -1
\]
\[
5x - y - 3 = -1
\]
\[
5x - y = 2 \quad \text{(Equation B)}
\]

### Step 3: Solve the new system of equations
Now, solve the system of equations (A) and (B):

1. \(x + y = 4\)
2. \(5x - y = 2\)

Add the two equations:
\[
(x + y) + (5x - y) = 4 + 2
\]
\[
6x = 6
\]
\[
x = 1
\]

Substitute \(x = 1\) into \(x + y = 4\):
\[
1 + y = 4
\]
\[
y = 3
\]

### Step 4: Find \(z\)
Substitute \(x = 1\) and \(y = 3\) into \(z = 2x + y - 3\):
\[
z = 2(1) + 3 - 3 = 2
\]

### Step 5: Calculate \(x + y + z\)
Now that we have \(x = 1\), \(y = 3\), and \(z = 2\), we can find \(x + y + z\):
\[
x + y + z = 1 + 3 + 2 = 6
\]

### Final Answer:
\[
x + y + z = 6
\]

Would you like further details or explanations on any step?

Here are 5 related questions you can explore:
1. How can substitution be used to solve more complex systems of equations?
2. What are other methods to solve systems of linear equations, such as matrix methods or elimination?
3. How can we use Cramer's rule to solve systems of equations?
4. What are the conditions for a system of equations to have no solution or infinitely many solutions?
5. How can we solve nonlinear systems of equations?

**Tip:** Always check your solutions by substituting them back into the original equations to ensure they satisfy all the equations.

Solve the system of equations: 2x + y - z = 3, x + 3y + 2z = 14, 3x - 2y + z = -1. What is x + y + z?

This page demonstrates how to solve a system of three linear equations using the substitution method. The system consists of 2x + y - z = 3, x + 3y + 2z = 14, and 3x - 2y + z = -1. Learn the step-by-step solution to find x, y, z, and x + y + z, suitable for Grades 9-12 students.

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