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To determine the values of \( k \) and \( h \) for which the given system of linear equations has infinitely many solutions, we need to check the conditions under which the system is dependent.

### The system of equations is:
\[
\begin{aligned}
4x - 5y - 5z &= 4 \\
7x + 9y - 3z &= 5 \\
-18x - 13y + hz &= k
\end{aligned}
\]

For a system to have infinitely many solutions, the third equation must be a linear combination of the first two equations. This implies that the coefficients of \( x \), \( y \), and \( z \) in the third equation must be proportional to the coefficients in the first two equations.

### Step-by-step process:

1. **Set up proportionality condition:**
   The third equation should be a combination of the first two. We write this as:
   \[
   (-18x - 13y + hz) = \alpha (4x - 5y - 5z) + \beta (7x + 9y - 3z)
   \]
   for some constants \( \alpha \) and \( \beta \).

2. **Match coefficients:**
   - For the \( x \)-terms:
     \[
     -18 = 4\alpha + 7\beta
     \]
   - For the \( y \)-terms:
     \[
     -13 = -5\alpha + 9\beta
     \]
   - For the \( z \)-terms:
     \[
     h = -5\alpha - 3\beta
     \]
   - For the constant terms (right-hand side):
     \[
     k = 4\alpha + 5\beta
     \]

3. **Solve the system of equations for \( \alpha \) and \( \beta \):**

   Using the first two equations:
   - From the \( x \)-term equation: \( 4\alpha + 7\beta = -18 \)
   - From the \( y \)-term equation: \( -5\alpha + 9\beta = -13 \)

   Solve these two simultaneous equations to find \( \alpha \) and \( \beta \).

4. **Calculate \( h \) and \( k \):**

   Once you have \( \alpha \) and \( \beta \), substitute them into the equations for \( h \) and \( k \) to determine their values.

### Solution:

After solving this system of equations, you should find that \( k = -14 \) and \( h = 11 \), which are the values that make the system have infinitely many solutions.

Would you like more details on how to solve the system step by step?

Here are 5 related questions you might be interested in:

1. How do you check if a system of linear equations is consistent or inconsistent?
2. What are the conditions for a system of equations to have a unique solution?
3. How does the rank of a matrix relate to the number of solutions in a system of linear equations?
4. Can you use matrix methods (like Gaussian elimination) to find \( k \) and \( h \) in this case?
5. What is the geometric interpretation of having infinitely many solutions?

**Tip:** A system of linear equations has infinitely many solutions when the equations represent the same plane or are dependent. This usually occurs when one equation is a linear combination of the others.

Learn how to determine the values of k and h in a system of linear equations to ensure infinitely many solutions. This problem covers key algebraic concepts such as linear systems, proportionality, and linear combinations. Suitable for students in Grades 10-12.

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