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To write the augmented matrix for the given system of equations, we first need to rearrange each equation in the standard form \( ax + by + cz = d \), where the variables \( x \), \( y \), and \( z \) are on the left side of the equation, and the constant term is on the right.

### Given System of Equations:
1. \(-3 - z = -4x\)
2. \(-3 = x - 5y\)
3. \(-3 + 2y + 5x = -z\)

### Step 1: Rewrite each equation in standard form

#### First Equation:
\[
-3 - z = -4x \Rightarrow 4x + 0y - z = -3
\]

#### Second Equation:
\[
-3 = x - 5y \Rightarrow -x + 5y + 0z = -3
\]

#### Third Equation:
\[
-3 + 2y + 5x = -z \Rightarrow 5x + 2y + z = 3
\]

Now, the system is:
1. \(4x + 0y - z = -3\)
2. \(-x + 5y + 0z = -3\)
3. \(5x + 2y + z = 3\)

### Step 2: Write the augmented matrix

The augmented matrix will take the form:

\[
\begin{bmatrix}
4 & 0 & -1 & | & -3 \\
-1 & 5 & 0 & | & -3 \\
5 & 2 & 1 & | & 3 \\
\end{bmatrix}
\]

This is the augmented matrix for the given system.

Would you like a breakdown of each step or further explanation?

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Here are five questions for further exploration:

1. How would you solve this system of equations using Gaussian elimination?
2. What is the significance of an augmented matrix in solving systems of linear equations?
3. Can you find the inverse of the matrix if it were square?
4. What are the next steps if we were to solve this system using matrix row operations?
5. How would you verify the solution to this system once you have it?

**Tip:** When constructing an augmented matrix, always ensure each equation is in standard form to avoid misplacing coefficients.

Write the augmented matrix for the following system of equations:
-3 - z = -4x
-3 = x - 5y
-3 + 2y + 5x = -z

Learn how to write the augmented matrix for a system of equations, including step-by-step rearrangement of equations into standard form and constructing the matrix. Suitable for high school algebra students.

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