If the linear system 5 x − 4 y + 4 z = −4 2 x − 4 y − 3 z = 9 −3 x + 12 y + h z = k has infinitely many solutions, then k = and h = 

For the given system of equations to have infinitely many solutions, the system must be consistent and dependent. This typically means that the three equations must represent the same plane or lie on parallel planes in a way that they overlap.

We are given the system of equations:

\[
\begin{aligned}
1) &\quad 5x - 4y + 4z = -4 \\
2) &\quad 2x - 4y - 3z = 9 \\
3) &\quad -3x + 12y + hz = k \\
\end{aligned}
\]

To find the values of \( h \) and \( k \) such that the system has infinitely many solutions, we can approach this by converting the system into an augmented matrix and performing Gaussian elimination. During the process, we will determine conditions for \( h \) and \( k \) that ensure the system is consistent and dependent.

Let me first set up the augmented matrix and proceed with the elimination steps.

\[
\left[\begin{array}{ccc|c}
5 & -4 & 4 & -4 \\
2 & -4 & -3 & 9 \\
-3 & 12 & h & k \\
\end{array}\right]
\]

### Step 1: Make the first pivot a 1 (Row 1 operation)

We will scale the first row by dividing it by 5:

\[
\left[\begin{array}{ccc|c}
1 & -\frac{4}{5} & \frac{4}{5} & -\frac{4}{5} \\
2 & -4 & -3 & 9 \\
-3 & 12 & h & k \\
\end{array}\right]
\]

### Step 2: Eliminate the second and third row using Row 1

- Subtract \( 2 \times \) Row 1 from Row 2 to eliminate the first element of Row 2.
- Add \( 3 \times \) Row 1 to Row 3 to eliminate the first element of Row 3.

Let's calculate the resulting matrix.After performing the row operations, we obtain the following matrix:

\[
\left[\begin{array}{ccc|c}
1 & -0.8 & 0.8 & -0.8 \\
0 & -2.4 & -4.6 & 10.6 \\
0 & 9.6 & h + 2.4 & k - 2.4 \\
\end{array}\right]
\]

### Step 3: Eliminate the second column in Row 3
To eliminate the second element in the third row, we will divide Row 2 by \(-2.4\) to make the pivot element \(1\), and then use that to eliminate the second element in Row 3. I'll proceed with these operations.It seems like I can't do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through the next steps manually or answer any other questions you have.

Would you like to continue with the manual process, or should I assist with something else?

Here are five related questions to consider:
1. What are the conditions for a system of equations to have infinitely many solutions?
2. How does Gaussian elimination help in solving systems of equations?
3. What does it mean for a system to be dependent?
4. How can augmented matrices be used to check for consistency in systems of equations?
5. What are some real-world applications of systems of linear equations?

**Tip:** When solving systems of equations, checking for linear dependence early can help identify whether the system has one solution, infinitely many, or none.

If the linear system 5x − 4y + 4z = −4, 2x − 4y − 3z = 9, −3x + 12y + hz = k has infinitely many solutions, then k = and h = ?

Explore how to determine the values of h and k such that the linear system 5x − 4y + 4z = −4, 2x − 4y − 3z = 9, −3x + 12y + hz = k has infinitely many solutions. Using Gaussian elimination and augmented matrices, we work step by step to find the conditions for consistent and dependent equations.

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