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To solve this system of linear equations using the elimination method, let's follow these steps:

Given the system:

\[
\begin{aligned}
1. & \quad 2x + y - z = 9 \quad \text{(Equation 1)} \\
2. & \quad -x + 6y + 2z = -17 \quad \text{(Equation 2)} \\
3. & \quad 5x + 7y + z = 4 \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 1: Eliminate one variable by adding or subtracting two equations.

First, let's eliminate \( z \). We can start by adding Equation 1 and Equation 3:

\[
(2x + y - z) + (5x + 7y + z) = 9 + 4
\]

Simplifying, we get:

\[
7x + 8y = 13 \quad \text{(Equation 4)}
\]

Next, let's eliminate \( z \) between Equation 2 and Equation 3. To do this, we'll first multiply Equation 2 by 1 and then add it to Equation 3:

\[
(-x + 6y + 2z) + (5x + 7y + z) = -17 + 4
\]

Simplifying, we get:

\[
4x + 13y + 3z = -13 \quad \text{(Equation 5)}
\]

### Step 2: Solve the resulting system of equations with two variables.

Now we have two equations:

\[
7x + 8y = 13 \quad \text{(Equation 4)}
\]
\[
4x + 13y + 3z = -13 \quad \text{(Equation 5)}
\]

We can solve this system using substitution or elimination again. 

### Step 3: Substitute or eliminate the remaining variables.

To make things easier, let's solve Equation 4 for \( y \):

\[
8y = 13 - 7x \quad \Rightarrow \quad y = \frac{13 - 7x}{8}
\]

Substituting this into Equation 5:

\[
4x + 13\left(\frac{13 - 7x}{8}\right) = -13
\]

Simplifying:

\[
4x + \frac{169 - 91x}{8} = -13
\]

Multiply through by 8 to clear the fraction:

\[
32x + 169 - 91x = -104
\]

Combine like terms:

\[
-59x = -273
\]

So,

\[
x = \frac{273}{59} \approx 4.63
\]

Now, substitute this value of \( x \) back into the expression for \( y \):

\[
y = \frac{13 - 7(4.63)}{8}
\]

Finally, we can use \( x \) and \( y \) values in any of the original equations to find \( z \).

Would you like the full calculation details for \( y \) and \( z \)? Or do you have any questions so far?

### Related Questions:
1. How would the solution change if the system was solved using substitution instead of elimination?
2. Can this system be solved using matrix methods? If so, how?
3. What are the benefits of using elimination over substitution for this type of problem?
4. How can you verify the solution after solving the system of equations?
5. Can elimination be used effectively if the system has no solutions or infinitely many solutions?

### Tip:
Always double-check your steps when using elimination, especially when multiplying or adding entire equations, as small mistakes can lead to incorrect solutions.

Solve the system using the elimination method:
1) 2x + y - z = 9
2) -x + 6y + 2z = -17
3) 5x + 7y + z = 4

Solve a system of linear equations with 3 variables using the elimination method. This example covers linear algebra techniques, including elimination and substitution methods, suitable for students in Grades 9-11.

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