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The problem is asking to construct the **augmented matrix** corresponding to the given system of equations. Here's how we solve it step-by-step.

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### Step 1: Write the given equations clearly
The system of equations is:
1. \( \frac{2x + 6y - z}{3} = 8 \)
2. \( 2(7z - 6x) + y - 9 = 3 \)
3. \( x - (5 + z) = 3y \)

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### Step 2: Rewrite the equations in standard form \( Ax + By + Cz = D \)
We rearrange each equation to isolate the terms involving \( x, y, z \) on one side and the constant on the other.

#### Equation 1:
\[
\frac{2x + 6y - z}{3} = 8
\]
Multiply through by 3:
\[
2x + 6y - z = 24
\]

#### Equation 2:
\[
2(7z - 6x) + y - 9 = 3
\]
Expand \( 2(7z - 6x) \):
\[
14z - 12x + y - 9 = 3
\]
Simplify:
\[
-12x + y + 14z = 12
\]

#### Equation 3:
\[
x - (5 + z) = 3y
\]
Distribute the negative sign:
\[
x - 5 - z = 3y
\]
Rearrange:
\[
x - 3y - z = 5
\]

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### Step 3: Form the augmented matrix
The augmented matrix includes the coefficients of \( x, y, z \) (in order) from each equation, and the constants on the right-hand side.

From the three equations:
1. \( 2x + 6y - z = 24 \)
2. \( -12x + y + 14z = 12 \)
3. \( x - 3y - z = 5 \)

The augmented matrix is:
\[
\begin{bmatrix}
2 & 6 & -1 & 24 \\
-12 & 1 & 14 & 12 \\
1 & -3 & -1 & 5
\end{bmatrix}
\]

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Would you like a step-by-step explanation of how to use this matrix for solving the system? Let me know! 

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### Follow-up Questions:
1. What is an augmented matrix, and why is it useful?
2. How can this matrix be used to solve the system using row-reduction?
3. What does it mean if the matrix has a row of all zeros after row-reduction?
4. Can this system be solved using determinants (Cramer's Rule)? If so, how?
5. How do you verify the solution obtained from the augmented matrix?

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### Tip:
When forming augmented matrices, ensure the variables \( x, y, z \) are consistently ordered across all equations for proper alignment!

Construct the augmented matrix corresponding to the following system of equations: 
1. (2x + 6y - z)/3 = 8
2. 2(7z - 6x) + y - 9 = 3
3. x - (5 + z) = 3y

Learn how to construct the augmented matrix for a system of linear equations. This problem involves rearranging equations into standard form and then creating the augmented matrix for the system. Suitable for high school students (Grades 9-11).

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