Math Problem Statement

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

Solution

We are given a system of three linear equations:

7x5y+4z=4(Equation 1)7x - 5y + 4z = 4 \quad \text{(Equation 1)} 5x+5y+8z=2(Equation 2)-5x + 5y + 8z = 2 \quad \text{(Equation 2)} 11x+5y+hz=k(Equation 3)-11x + 5y + hz = k \quad \text{(Equation 3)}

We need to find values of kk and hh 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 hh and kk.

Step 1: Eliminate terms

Start by eliminating one variable, say xx, from Equation 2 and Equation 3 by combining them with Equation 1. This will allow us to identify constraints on hh and kk.

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

h=kz+20y44z+447zh = \frac{k}{z} + \frac{20y - 44z + 44}{7z}

To achieve infinitely many solutions, hh and kk must satisfy this relation. However, since this system involves constraints based on other variables, we'll need further clarification if specific values of yy or zz 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 hh and kk 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 hh and kk, you can use matrix methods like the determinant to check for infinite or no solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Gaussian Elimination
Infinitely Many Solutions

Formulas

Linear equation system in 3 variables
Gaussian Elimination method for solving systems

Theorems

Linear Dependency Condition
Conditions for Infinite Solutions

Suitable Grade Level

Undergraduate (Linear Algebra or Advanced High School)