https://file.aiquickdraw.com/mathbot/file/1726422395214wlgxa73s.jpg

We are given a system of linear equations with variables \(x\), \(y\), and \(z\):

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

To find the values of \(k\) and \(h\) such that the system has infinitely many solutions, we need to analyze the consistency of the system.

### Step-by-step method:

1. **Convert to matrix form**: We can represent this system as an augmented matrix and then perform row reduction to see under what conditions the system has infinitely many solutions.
   The augmented matrix is:

   \[
   \left[
   \begin{array}{ccc|c}
   6 & -3 & 5 & -7 \\
   -4 & 4 & 4 & -2 \\
   10 & -13 & h & k
   \end{array}
   \right]
   \]

2. **Row operations**: We will perform Gaussian elimination on the matrix and analyze the conditions for \(k\) and \(h\) to make the system consistent and dependent (i.e., have infinitely many solutions). This requires finding conditions where the system reduces to a row of zeros, indicating that one equation is a combination of the others.

I will now calculate this and check the conditions.After performing row reduction, the augmented matrix in reduced row echelon form (RREF) is:

\[
\begin{pmatrix}
1 & 0 & 0 & \frac{-17h - 16k - 117}{6h + 126} \\
0 & 1 & 0 & \frac{-10h - 11k - 45}{3h + 63} \\
0 & 0 & 1 & \frac{k - 15}{h + 21}
\end{pmatrix}
\]

### For the system to have infinitely many solutions:
1. **The denominator cannot be zero**, so:
   \[
   6h + 126 \neq 0, \quad 3h + 63 \neq 0, \quad h + 21 \neq 0
   \]
   This simplifies to \( h \neq -21 \).

2. **For infinitely many solutions, the system must be consistent and have at least one dependent equation**. This occurs when the right-hand side of the third row becomes 0, meaning:
   \[
   \frac{k - 15}{h + 21} = 0
   \]
   This implies \( k = 15 \).

Thus, \( k = 15 \) and \( h \neq -21 \) must hold for the system to have infinitely many solutions. Specifically, the system is dependent for these values.

### Final values:
- \( k = 15 \)
- \( h = -21 \)

Would you like a further breakdown of any steps or have additional questions?

### Related questions:
1. What are the conditions for a system of linear equations to have no solutions?
2. How does Gaussian elimination help in solving linear systems?
3. Can we find the solution set for specific values of \(k\) and \(h\)?
4. What are the implications of having a row of zeros in a matrix?
5. What is the difference between dependent and independent systems in linear algebra?

### Tip:
When solving systems of equations, always ensure that the matrix operations maintain consistency across all rows to avoid introducing errors during elimination.

If the linear system has infinitely many solutions, then k = ___ and h = ___.

Learn how to determine the values of k and h that make the given system of linear equations have infinitely many solutions. The problem involves concepts from linear algebra, Gaussian elimination, and conditions for infinite solutions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation