https://file.aiquickdraw.com/mathbot/file/1728174212287y1uqlp6s.jpg

To construct the augmented matrix corresponding to the system of equations in the image, let’s break down the given system:

1. \( \frac{7 + 6x}{7} = y \)
2. \( 7z - 4(x - 5y) = 0 \)
3. \( 5x - y = 2(x - 3z) \)

### Step 1: Simplify the system
We'll simplify each equation.

#### Equation 1:
\[
\frac{7 + 6x}{7} = y \quad \Rightarrow \quad 7 + 6x = 7y
\]
Thus, the equation becomes:
\[
6x - 7y = -7
\]

#### Equation 2:
\[
7z - 4(x - 5y) = 0 \quad \Rightarrow \quad 7z - 4x + 20y = 0
\]
Thus, the equation becomes:
\[
-4x + 20y + 7z = 0
\]

#### Equation 3:
\[
5x - y = 2(x - 3z) \quad \Rightarrow \quad 5x - y = 2x - 6z
\]
Thus, the equation becomes:
\[
3x - y + 6z = 0
\]

### Step 2: Write the augmented matrix
The system of equations now is:

1. \( 6x - 7y + 0z = -7 \)
2. \( -4x + 20y + 7z = 0 \)
3. \( 3x - y + 6z = 0 \)

The augmented matrix is:
\[
\begin{pmatrix}
6 & -7 & 0 & | & -7 \\
-4 & 20 & 7 & | & 0 \\
3 & -1 & 6 & | & 0
\end{pmatrix}
\]

Let me know if you'd like more details on any step!

### Further questions you could explore:
1. How do you perform row reduction to solve the augmented matrix?
2. What is the geometric interpretation of the solutions to this system?
3. How can you solve a system of equations using matrix inversion?
4. What is the difference between homogeneous and non-homogeneous systems of equations?
5. How would the system change if one of the equations was inconsistent?

### Tip:
When constructing augmented matrices, make sure that the coefficients are aligned correctly and remember to include the constants in the last column.

Construct the augmented matrix that corresponds to the following system of equations: (7 + 6x)/7 = y, 7z - 4(x - 5y) = 0, 5x - y = 2(x - 3z)

Learn how to construct an augmented matrix for a system of equations. This problem simplifies algebraic expressions and involves Gaussian elimination techniques, suitable for high school or college students.

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