Math Problem Statement
Consider the following system of equations. { 2π₯β3π¦+3π§=2π π₯β2π¦+4π§=5π βπ¦+5π§=8π2 Determine the following. Assume that π is a real number, a. The values of π, if any, that would result in the system of equations having a unique solution.
Solution
To determine the values of that result in a unique solution, we need to check the determinant of the coefficient matrix. If the determinant is nonzero, the system has a unique solution.
Step 1: Write the System as a Matrix Equation
The given system of equations is:
2x - 3y + 3z &= 2k \\ x - 2y + 4z &= 5k \\ - y + 5z &= 8k^2 \end{aligned}$$ The coefficient matrix $$A$$ is: $$A = \begin{bmatrix} 2 & -3 & 3 \\ 1 & -2 & 4 \\ 0 & -1 & 5 \end{bmatrix}$$ The determinant of $$A$$ must be nonzero for the system to have a unique solution. ### Step 2: Compute the Determinant of $$A$$ $$\det(A) = \begin{vmatrix} 2 & -3 & 3 \\ 1 & -2 & 4 \\ 0 & -1 & 5 \end{vmatrix}$$ Expanding along the first column: $$\det(A) = 2 \begin{vmatrix} -2 & 4 \\ -1 & 5 \end{vmatrix} - (-3) \begin{vmatrix} 1 & 4 \\ 0 & 5 \end{vmatrix} + 3 \begin{vmatrix} 1 & -2 \\ 0 & -1 \end{vmatrix}$$ Now, compute the 2Γ2 determinants: $$\begin{vmatrix} -2 & 4 \\ -1 & 5 \end{vmatrix} = (-2)(5) - (4)(-1) = -10 + 4 = -6$$ $$\begin{vmatrix} 1 & 4 \\ 0 & 5 \end{vmatrix} = (1)(5) - (4)(0) = 5$$ $$\begin{vmatrix} 1 & -2 \\ 0 & -1 \end{vmatrix} = (1)(-1) - (-2)(0) = -1$$ Substituting these values: $$\det(A) = 2(-6) + 3(5) + 3(-1)$$ $$= -12 + 15 - 3 = 0$$ Since $$\det(A) = 0$$, the system does **not** have a unique solution for **any** value of $$k$$. Instead, it may have **infinitely many solutions or no solution**, depending on the consistency of the augmented matrix. ### Conclusion: There is **no value of $$k$$** that results in a unique solution, because the determinant of the coefficient matrix is always **zero**. --- Would you like to analyze whether the system has **infinitely many solutions or no solution** based on $$k$$? Let me know! Here are some related questions to explore: 1. What happens when the determinant of a matrix is zero? 2. How do we determine if a system has infinitely many solutions? 3. How do we check for inconsistency in a system of equations? 4. What is the rank of a matrix, and how does it relate to solutions? 5. What is Cramer's Rule, and why doesn't it apply when the determinant is zero? **Tip:** The rank of a matrix helps determine if a system has infinitely many solutions or no solution. If the rank of the coefficient matrix equals the rank of the augmented matrix, the system is consistent!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Determinants
Formulas
Determinant of a 3x3 matrix
Determinant condition for unique solution
Theorems
Cramer's Rule
Rank of a matrix
Determinant of coefficient matrix
Suitable Grade Level
Grades 10-12
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