https://file.aiquickdraw.com/mathbot/file/1726447076944wf0xkcj4.jpg

We are given a system of three linear equations:

\[
7x - 5y + 4z = 4 \quad \text{(Equation 1)}
\]
\[
-5x + 5y + 8z = 2 \quad \text{(Equation 2)}
\]
\[
-11x + 5y + hz = k \quad \text{(Equation 3)}
\]

We need to find values of \( k \) and \( h \) such that this system has **infinitely many solutions**.

### Steps to solve:

1. **Condition for infinite solutions**: For a system to have infinitely many solutions, the three equations must be linearly dependent, meaning one equation must be a linear combination of the others.

2. We will use **Gaussian elimination** to reduce the system and find relationships between \( h \) and \( k \).

#### Step 1: Eliminate terms

Start by eliminating one variable, say \( x \), from Equation 2 and Equation 3 by combining them with Equation 1. This will allow us to identify constraints on \( h \) and \( k \).

I'll now proceed with the calculations and find the values of \( k \) and \( h \).After performing the elimination process, the condition for \( h \) and \( k \) to ensure infinitely many solutions is derived as:

\[
h = \frac{k}{z} + \frac{20y - 44z + 44}{7z}
\]

To achieve infinitely many solutions, \( h \) and \( k \) must satisfy this relation. However, since this system involves constraints based on other variables, we'll need further clarification if specific values of \( y \) or \( z \) are expected, or alternatively, you can provide additional context (like simplifying assumptions).

Do you want further details or clarifications?

Here are some related questions:
1. What is the significance of the linear dependency condition for infinite solutions?
2. How does Gaussian elimination help simplify systems of linear equations?
3. What happens if there are no values of \( h \) and \( k \) that satisfy the system?
4. What are other ways to check if a system has infinitely many solutions?
5. How would the solution change if the system had a unique solution instead of infinitely many?

**Tip**: For systems with parameters like \( h \) and \( k \), you can use matrix methods like the determinant to check for infinite or no solutions.

If the linear system has infinitely many solutions, then k = ___ and h = ___: 7x - 5y + 4z = 4, -5x + 5y + 8z = 2, -11x + 5y + hz = k.

Solve for k and h in a system of three linear equations using Gaussian Elimination. Understand the conditions for infinitely many solutions through linear dependency and algebraic manipulation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation