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

7+4x7=y2z−2(x−6y)=05x−y=4(x−6z)




The given system of equations appears to involve three variables, \(x\), \(y\), and \(z\). Let's rewrite the equations in a clear, standard form to identify the coefficients:

### Step 1: Rewrite the system of equations

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

### Step 2: Simplify the equations

1. **First equation**: \( 7 + 4x = y + 2z \)
   \[
   4x - y - 2z = -7
   \]
   
2. **Second equation**: \( -2(x - 6y) = 0 \)
   \[
   -2x + 12y = 0 \quad \text{(or)} \quad x - 6y = 0
   \]

3. **Third equation**: \( 5x - y = 4(x - 6z) \)
   \[
   5x - y = 4x - 24z
   \]
   \[
   x - y + 24z = 0
   \]

### Step 3: Write the system in matrix form

The system of equations can now be written as:
1. \( 4x - y - 2z = -7 \)
2. \( x - 6y = 0 \)
3. \( x - y + 24z = 0 \)

We can now represent this system as an augmented matrix:

\[
\begin{pmatrix}
4 & -1 & -2 & | & -7 \\
1 & -6 & 0 & | & 0 \\
1 & -1 & 24 & | & 0
\end{pmatrix}
\]

This is the augmented matrix corresponding to the system of equations.

Would you like further details or have any questions about this solution?

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Here are 5 related questions:
1. How do you solve a system of equations using the augmented matrix method?
2. What are the possible outcomes of solving a system using matrices (unique solution, no solution, infinitely many solutions)?
3. How do you perform Gaussian elimination on an augmented matrix?
4. How can the determinant of a matrix help in solving a system of linear equations?
5. How do you interpret the solutions of a system when the augmented matrix has a row of all zeros?

**Tip**: When working with matrices, keep track of your row operations carefully to avoid introducing errors.

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

Learn how to construct the augmented matrix for a system of linear equations involving three variables (x, y, z). This step-by-step solution covers simplifying equations and matrix representation, using key concepts in linear algebra. Suitable for high school students in Grades 10-12.

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