Math Problem Statement
Solution
To solve the given system of equations and determine the condition for consistency, we need to ensure that the determinant of the coefficient matrix is non-zero. This guarantees that the system has a unique solution, making it consistent.
The system of equations provided is:
Step 1: Coefficient Matrix
The coefficient matrix of the system is:
6 & -3 & -7 \\ 24 & 5 & k \\ 9 & 4 & -3 \end{pmatrix}$$ For the system to be consistent, the determinant of the matrix must not be zero. Let’s compute the determinant of $$A$$. $$\det(A) = \begin{vmatrix} 6 & -3 & -7 \\ 24 & 5 & k \\ 9 & 4 & -3 \end{vmatrix}$$ We can expand this determinant along the first row: $$\det(A) = 6 \begin{vmatrix} 5 & k \\ 4 & -3 \end{vmatrix} - (-3) \begin{vmatrix} 24 & k \\ 9 & -3 \end{vmatrix} + (-7) \begin{vmatrix} 24 & 5 \\ 9 & 4 \end{vmatrix}$$ Now I will compute the determinant for each 2x2 matrix. Let me calculate that for you:The determinant of the matrix $$A$$ is: $$\det(A) = -51k - 663$$ ### Step 2: Condition for Consistency For the system to be consistent, the determinant must not be equal to zero: $$-51k - 663 \neq 0$$ Solving for $$k$$, we get: $$k \neq -\frac{663}{51} = -13$$ Thus, the system will be consistent as long as $$k \neq -13$$. Would you like further clarification on any part of this solution? Here are some related questions for further exploration: 1. How do you solve a system of equations using matrix methods like Cramer's Rule? 2. What does it mean for a system of equations to be consistent or inconsistent? 3. Can you explain how to find the inverse of a matrix when the determinant is non-zero? 4. How do you solve a system of linear equations using Gaussian elimination? 5. What are the applications of determinants in linear algebra? **Tip**: Always check the determinant of a matrix when determining if a system of linear equations has a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Determinants
Formulas
Determinant of a 3x3 matrix
Expansion of determinant along a row
Consistency condition for systems of linear equations
Theorems
Determinants and Invertibility
System Consistency Theorem
Suitable Grade Level
Grades 11-12 or Early College Level
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