Solve the system of equations by elimination.
  x − 2y + 3z = 9
−x + 3y – z = −16   
2x – 5y + 5z = 17


To solve the system of equations using the elimination method, let's proceed step by step. Here are the equations:

\[
\begin{aligned}
1) & \quad x - 2y + 3z = 9 \\
2) & \quad -x + 3y - z = -16 \\
3) & \quad 2x - 5y + 5z = 17
\end{aligned}
\]

### Step 1: Eliminate \(x\)

We'll add equations 1 and 2 to eliminate \(x\) and then work with equation 3.

1. Add equation 1 and equation 2:

\[
(x - 2y + 3z) + (-x + 3y - z) = 9 + (-16)
\]

Simplifying this:

\[
y + 2z = -7 \quad \text{(Equation 4)}
\]

2. Now, we'll work with equation 3 and equation 1. Multiply equation 1 by 2 and then subtract it from equation 3:

\[
2(x - 2y + 3z) = 2(9)
\]

This gives:

\[
2x - 4y + 6z = 18 \quad \text{(Equation 5)}
\]

Subtract equation 5 from equation 3:

\[
(2x - 5y + 5z) - (2x - 4y + 6z) = 17 - 18
\]

Simplifying this:

\[
-y - z = -1 \quad \text{(Equation 6)}
\]

### Step 2: Solve for \(y\) and \(z\)

Now, we have two equations in terms of \(y\) and \(z\):

\[
\begin{aligned}
4) & \quad y + 2z = -7 \\
6) & \quad -y - z = -1
\end{aligned}
\]

Add equation 4 and equation 6:

\[
(y + 2z) + (-y

Learn how to solve a system of equations using the elimination method with a detailed step-by-step guide. This problem involves three linear equations with variables x, y, and z. Suitable for high school students studying algebra and linear equations.

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