https://file.aiquickdraw.com/mathbot/file/1726421846908pr2loqqb.jpg

The system of linear equations presented in the image is:

\[
\begin{aligned}
6x - 3y + 5z &= -7 \\
-4x + 4y + 4z &= -2 \\
10x - 13y + hz &= k
\end{aligned}
\]

We are tasked with finding values of \( k \) and \( h \) such that the system has infinitely many solutions. 

### General Condition for Infinitely Many Solutions:
A system of linear equations will have infinitely many solutions if the equations are dependent, meaning that one of the equations can be written as a linear combination of the others. To achieve this, the system must have consistent row-reduction results, and the last row of the augmented matrix must result in a row of zeros after Gaussian elimination.

### Steps to Solve:

1. **Convert the system into an augmented matrix:**

\[
\begin{bmatrix}
6 & -3 & 5 & | & -7 \\
-4 & 4 & 4 & | & -2 \\
10 & -13 & h & | & k
\end{bmatrix}
\]

2. **Perform Gaussian elimination** to identify relationships between \( k \) and \( h \).

Let's start row reducing to find conditions for \( k \) and \( h \).After performing Gaussian elimination on the system, the augmented matrix has been reduced to:

\[
\begin{bmatrix}
1 & -0.5 & 0.833 & -1.167 \\
0 & 1 & 3.667 & -3.333 \\
0 & 0 & 1 & -0.714
\end{bmatrix}
\]

### Analysis of the Result:
- The matrix is in row-echelon form, and each row has a leading 1, which implies that the system does not currently have infinitely many solutions. For the system to have infinitely many solutions, one of the rows should become all zeros. This happens when specific values of \( h \) and \( k \) make one of the rows dependent.

To achieve this, we will examine the conditions where the last row becomes zero by determining the relationships between \( h \) and \( k \). Let me now find those conditions.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like help solving this manually? I can guide you through the steps, or provide explanations for other concepts.

Here are 5 related questions:
1. How do you find the rank of a matrix from its row echelon form?
2. What is the difference between Gaussian elimination and Gauss-Jordan elimination?
3. How do you determine when a system of equations has no solution, one solution, or infinitely many solutions?
4. What are the steps to solving a system of equations using matrices?
5. How can parametric solutions be used when solving systems with infinitely many solutions?

**Tip:** Always check for dependent rows when analyzing linear systems to identify cases of infinitely many solutions!

If the linear system 6x - 3y + 5z = -7, -4x + 4y + 4z = -2, and 10x - 13y + hz = k has infinitely many solutions, then what are the values of k and h?

Discover how to find the values of k and h that make the system of equations 6x - 3y + 5z = -7, -4x + 4y + 4z = -2, and 10x - 13y + hz = k have infinitely many solutions. This problem uses concepts from linear algebra, including Gaussian elimination and the augmented matrix method.

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