Solve the system of equations by elimination.
2x–y–z=
–
6
3x–3y+z=
–
1
–
3x+3y+z=17

Let's solve the given system of equations using the elimination method. The system of equations is:

\[
\begin{aligned}
1) & \quad 2x - y - z = -6 \\
2) & \quad 3x - 3y + z = -1 \\
3) & \quad -3x + 3y + z = 17
\end{aligned}
\]

### Step 1: Add Equation 2 and Equation 3

First, let's add equations 2 and 3 to eliminate \(x\):

\[
(3x - 3y + z) + (-3x + 3y + z) = -1 + 17
\]

This simplifies to:

\[
2z = 16 \quad \Rightarrow \quad z = 8
\]

### Step 2: Substitute \(z = 8\) into the original equations

Now, substitute \(z = 8\) back into the first and second original equations to find \(x\) and \(y\).

Substituting into equation 1:

\[
2x - y - 8 = -6 \quad \Rightarrow \quad 2x - y = 2 \quad \text{(Equation 4)}
\]

Substituting into equation 2:

\[
3x - 3y + 8 = -1 \quad \Rightarrow \quad 3x - 3y = -9 \quad \Rightarrow \quad x - y = -3 \quad \text{(Equation 5)}
\]

### Step 3: Solve for \(x\) and \(y\) using Equation 4 and Equation 5

We have the system:

\[
\begin{aligned}
2x - y &= 2 \quad \text{(Equation 4)}\\
x - y &= -3 \quad \text{(Equation 5)}
\end{aligned}
\]

Subtract Equation 5 from Equation 4:

\[
(2x - y) - (x - y) = 2 - (-3)
\]

Simplifying:

\[
x = 5
\]

Substitute \(x = 5\) back into Equation 5:

\[
5 - y = -3 \quad \Rightarrow \quad y = 8
\]

### Final Solution
The solution to the system of equations is:
\[
\boxed{x = 5, \, y = 8, \, z = 8}
\]

Would you like more details or have any questions?

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**Related Questions:**

1. How do you solve systems of equations using substitution?
2. What is the graphical interpretation of solving systems of equations?
3. How can you verify the solution to a system of equations?
4. What are the conditions for a system of linear equations to have no solution?
5. How can matrices be used to solve systems of equations?

**Tip:** Always double-check your work by substituting the solutions back into the original equations to verify they satisfy all equations.

Solve the system of equations by elimination: 2x - y - z = -6, 3x - 3y + z = -1, -3x + 3y + z = 17.

Learn how to solve a system of linear equations using the elimination method. This example includes the equations 2x - y - z = -6, 3x - 3y + z = -1, and -3x + 3y + z = 17. The solution involves eliminating variables to find the values of x, y, and z. Suitable for high school students in Grades 9-12.

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