Math Problem Statement
If the linear system has infinitely many solutions, then k = ___ and h = ___.
Solution
We are given a system of linear equations with variables , , and :
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems
Gaussian Elimination
Infinitely Many Solutions
Formulas
Augmented matrix representation of the system
Row reduction using Gaussian elimination
Theorems
Conditions for a system of equations to have infinitely many solutions
Consistency of linear systems
Suitable Grade Level
Grades 10-12
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