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For **Question 4**, you are asked to write the given system of equations as an augmented matrix. The system provided is:

\[
\begin{aligned}
7x - y &= 12 \\
3x + y &= 8
\end{aligned}
\]

### Writing this as an augmented matrix:

The coefficients of the variables and the constants from the right-hand side of the equations will be placed in the matrix:

\[
\begin{bmatrix}
7 & -1 & | & 12 \\
3 &  1 & | & 8
\end{bmatrix}
\]

This matrix represents the system of equations. The first column corresponds to \(x\), the second column corresponds to \(y\), and the last column after the vertical bar represents the constants on the right-hand side of the equations.

---

For **Question 5**, you are asked to convert the given augmented matrix into a system of equations. The augmented matrix provided is:

\[
\begin{bmatrix}
6 & -4 & -3 & | & -44 \\
-1 & -5 & 2 & | & -18 \\
4 & 3 & 5 & | & 13
\end{bmatrix}
\]

Each row in this matrix represents a linear equation. The first three columns represent the coefficients of \(x\), \(y\), and \(z\), respectively, and the last column represents the constants on the right-hand side. Thus, the corresponding system of equations is:

1. \( 6x - 4y - 3z = -44 \)
2. \( -x - 5y + 2z = -18 \)
3. \( 4x + 3y + 5z = 13 \)

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Would you like further details or clarifications? Let me know if you need any help filling in the blanks.

Here are five additional related questions:
1. How do you solve an augmented matrix using Gaussian elimination?
2. What are the conditions for a system of equations to have a unique solution, no solution, or infinitely many solutions?
3. How can matrix row operations simplify solving a system of equations?
4. What is the significance of the determinant when solving systems of linear equations?
5. How can we determine if a matrix is invertible, and what role does it play in solving equations?

**Tip:** When solving systems of equations with matrices, try to identify patterns in the coefficients to speed up the process of creating the augmented matrix.

Write the system of equations as an augmented matrix: 
7x - y = 12 
3x + y = 8

Write the system of equations that corresponds to the augmented matrix: 
[ 6 -4 -3 | -44 ] 
[-1 -5 2 | -18 ] 
[ 4 3 5 | 13 ]

Learn how to convert systems of linear equations into augmented matrices and solve them step by step. This guide covers two key problems, including converting a system of two equations and an augmented matrix. Suitable for Grades 10-12.

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