https://file.aiquickdraw.com/mathbot/file/17278431008103h2xizdv.jpg

Let's break down the system of equations from the image step by step so we can construct the augmented matrix.

The system of equations given is:

\[
\begin{aligned}
1. \ \frac{5 + 7x}{5} &= y \\
2. \ 5z - 3(x - 6y) &= 0 \\
3. \ 5x - y &= 2(x - 6z)
\end{aligned}
\]

### Step 1: Rewrite all equations in standard form.

1. First equation:
\[
\frac{5 + 7x}{5} = y \quad \text{(Multiply both sides by 5 to eliminate the fraction)}
\]
\[
5 + 7x = 5y \quad \Rightarrow \quad 7x - 5y = -5
\]

2. Second equation:
\[
5z - 3(x - 6y) = 0 \quad \text{(Expand)}
\]
\[
5z - 3x + 18y = 0 \quad \Rightarrow \quad -3x + 18y + 5z = 0
\]

3. Third equation:
\[
5x - y = 2(x - 6z) \quad \text{(Expand the right-hand side)}
\]
\[
5x - y = 2x - 12z \quad \Rightarrow \quad 5x - y - 2x + 12z = 0
\]
\[
3x - y + 12z = 0
\]

### Step 2: Augmented Matrix Representation

Now that we have the system of equations in standard form:

\[
\begin{aligned}
7x - 5y + 0z &= -5 \\
-3x + 18y + 5z &= 0 \\
3x - y + 12z &= 0
\end{aligned}
\]

The augmented matrix will be:

\[
\begin{bmatrix}
7 & -5 & 0 & -5 \\
-3 & 18 & 5 & 0 \\
3 & -1 & 12 & 0
\end{bmatrix}
\]

This matrix corresponds to the system of equations, where each row represents an equation, and the final column represents the constants on the right-hand side of the equations.

Would you like any further details or have any questions?

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Here are 5 related questions to explore:

1. How would you solve this system of equations using Gaussian elimination?
2. What is the significance of the augmented matrix in solving systems of linear equations?
3. How can you find the determinant of the coefficient matrix for this system?
4. Can you use Cramer's rule to solve this system, and if so, how?
5. How does this system of equations relate to linear algebra and vector spaces?

**Tip:** When solving systems of linear equations, always rewrite them in a consistent format (standard form) before forming the augmented matrix. This ensures clarity and accuracy during the solving process.

Construct the augmented matrix that corresponds to the following system of equations:

(5 + 7x)/5 = y
5z - 3(x - 6y) = 0
5x - y = 2(x - 6z)

Learn how to construct an augmented matrix for a system of linear equations. This example covers the transformation of the system into standard form and the creation of the augmented matrix, applying key concepts from linear algebra. Suitable for undergraduate students.

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