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To find the missing number in the reduced row echelon form (RREF) matrix, let's analyze the augmented matrix and how the transformation to RREF is carried out.

The original augmented matrix is:

\[
\begin{bmatrix}
4 & -4 & 9 & | & -12 \\
1 & 1 & 1 & | & -\frac{17}{6} \\
-9 & -14 & -6 & | & 25
\end{bmatrix}
\]

The RREF form given is:

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

### Step-by-Step Solution

In the RREF form, each row corresponds to solving for a single variable. Here:

1. The first row indicates that \( x = -4 \).
2. The third row gives \( z = \frac{2}{3} \).
3. The second row represents \( y = ? \), which we need to determine.

### Method to Find the Missing Value

To find the missing entry, we can substitute the known values of \( x \) and \( z \) back into the second equation from the original matrix, solve for \( y \), and thus complete the transformation.

Using the second row in the original matrix:

\[
1 \cdot x + 1 \cdot y + 1 \cdot z = -\frac{17}{6}
\]

Substitute \( x = -4 \) and \( z = \frac{2}{3} \):

\[
-4 + y + \frac{2}{3} = -\frac{17}{6}
\]

To solve for \( y \), first combine the constants on the left-hand side:

\[
y = -\frac{17}{6} + 4 - \frac{2}{3}
\]

Convert \( 4 \) and \( -\frac{2}{3} \) to fractions with a denominator of 6:

\[
y = -\frac{17}{6} + \frac{24}{6} - \frac{4}{6} = \frac{3}{6} = \frac{1}{2}
\]

### Answer

The missing number is \( \frac{1}{2} \).

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Would you like further explanation on any of the steps? 

Here are some related questions:
1. How is the reduced row echelon form calculated?
2. What are the rules for transforming matrices to RREF?
3. How do we interpret an augmented matrix?
4. Why do we need RREF in solving linear equations?
5. How does substitution work in matrix row transformations?

**Tip:** Always verify each variable in RREF by substituting back into the original equations to confirm correctness.

A system of equations is converted to an augmented matrix, which is then converted to reduced row echelon form. Find the missing number in the reduced row echelon form matrix.

Learn how to find missing values in the reduced row echelon form (RREF) of an augmented matrix derived from a system of equations. This solution explains matrix row operations and substitution methods, suitable for high school students in Grades 10-12.

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