I have the following equations -2x+y+6z=-4 and -x+y+4z=-3 what is the redcuded\echelon form of this

To find the reduced row echelon form (RREF) of the system of equations, we'll first represent the system as an augmented matrix, and then apply Gaussian elimination (row reduction) to transform it into RREF.

The given system of equations is:

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

### Step 1: Write the augmented matrix

\[
\begin{pmatrix}
-2 & 1 & 6 & | & -4 \\
-1 & 1 & 4 & | & -3
\end{pmatrix}
\]

### Step 2: Perform row operations

#### Row 1: Multiply Row 2 by 2 and subtract it from Row 1 to eliminate the first element of Row 2

- Multiply Row 2 by \(2\): \([ -2, 2, 8, -6 ]\)
- Subtract this from Row 1:
  
  Row 1: \( [ -2, 1, 6, -4 ] - [ -2, 2, 8, -6 ] = [ 0, -1, -2, 2 ] \)

This results in the matrix:
\[
\begin{pmatrix}
0 & -1 & -2 & | & 2 \\
-1 & 1 & 4 & | & -3
\end{pmatrix}
\]

#### Row 3: Swap rows to bring the non-zero pivot to the top
\[
\begin{pmatrix}
-1 & 1 & 4 & | & -3 \\
0 & -1 & -2 & | & 2
\end{pmatrix}
\]

### Step 3: Make leading coefficient of the first row 1
- Multiply Row 1 by \(-1\):

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

### Step 4: Eliminate the second pivot element
- Multiply Row 2 by \(-1\):

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

- Add Row 2 to Row 1 to eliminate the second pivot:
  \([ 1, -1, -4, 3 ] + [ 0, 1, 2, -2 ] = [ 1, 0, -2, 1 ]\)

Thus, the reduced row echelon form is:

\[
\begin{pmatrix}
1 & 0 & -2 & | & 1 \\
0 & 1 & 2 & | & -2
\end{pmatrix}
\]

### Final Solution:
This matrix corresponds to the system of equations:
1. \( x - 2z = 1 \)
2. \( y + 2z = -2 \)

This gives the parametric solution:
- \( x = 1 + 2z \)
- \( y = -2 - 2z \)

Would you like any further clarification?

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Here are 5 related questions:
1. How do you interpret the solution in terms of a geometric plane?
2. How does Gaussian elimination differ from Gauss-Jordan elimination?
3. What does it mean if a system of equations has no solution in terms of row operations?
4. How can row echelon form be used to determine the rank of a matrix?
5. Can Gaussian elimination be used for any system of linear equations?

**Tip:** The number of free variables corresponds to the number of non-pivot columns in the RREF.

I have the following equations -2x+y+6z=-4 and -x+y+4z=-3 what is the reduced row echelon form of this

Learn how to find the reduced row echelon form (RREF) for the system of equations -2x + y + 6z = -4 and -x + y + 4z = -3 using Gaussian elimination. This problem demonstrates key matrix operations and transformations, and is suitable for students in Grades 10-12.

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