Math Problem Statement
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Solution
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We are given a system of linear equations in the form of a matrix:
3 & -1 & 2 \\ -3 & 3 + 8a & -6 - 4b \\ \end{pmatrix}$$ **Part a**: We need to determine for which value of $$a$$ the system has no solutions, given that $$b \neq -1$$. ### Condition for no solution: A system of equations will have no solutions if the determinant of the coefficient matrix is zero, leading to inconsistent equations. Let's compute the determinant of the matrix and solve for $$a$$ under this condition. Matrix: $$\begin{pmatrix} 3 & -1 \\ -3 & 3 + 8a \\ \end{pmatrix}$$ ### Determinant of this 2x2 matrix: $$\text{Det} = 3(3 + 8a) - (-1)(-3) = 9 + 24a - 3 = 6 + 24a$$ For no solutions, the determinant must be zero: $$6 + 24a = 0 \implies a = -\frac{6}{24} = -\frac{1}{4}$$ So, $$a = -\frac{1}{4}$$ when the system has no solutions, provided that $$b \neq -1$$. **Part b**: The system has a unique solution for all values of $$a$$ except one. This value of $$a$$ corresponds to the case where the determinant is zero, which we already found in part (a). Hence, the value is: $$a = -\frac{1}{4}$$ **Part c**: The system has infinitely many solutions when both the determinant is zero and the system is consistent. For consistency, the second equation must be a scalar multiple of the first. We already know that for no solutions, $$a = -\frac{1}{4}$$. Now, we need to check when $$b = -1$$, since that would allow the system to have infinitely many solutions due to linear dependency. Thus, for infinitely many solutions, we have: $$a = -\frac{1}{4}, b = -1$$ ### Final answers: - **a)** $$a = -\frac{1}{4}$$ - **b)** $$a = -\frac{1}{4}$$ - **c)** $$a = -\frac{1}{4}, b = -1$$ Let me know if you need further clarifications or have more questions! Here's a tip: always check for both determinant and consistency when solving linear systems. #### Related Questions: 1. How do you calculate the determinant of a 2x2 matrix? 2. What does it mean for a system of equations to have no solution geometrically? 3. How do you find the condition for infinitely many solutions in a system of linear equations? 4. What role does the determinant play in determining the number of solutions? 5. How does changing a parameter in a matrix affect the solution of a system? **Tip**: When solving linear systems, always try to simplify the matrix (if possible) using row operations to make the calculations easier.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Determinants
Matrix Theory
Formulas
Determinant of 2x2 matrix: det(A) = ad - bc
Condition for no solutions: det(A) = 0
Consistency conditions for systems of equations
Theorems
Determinant Theorem
Unique Solution Theorem for Systems of Equations
Suitable Grade Level
Undergraduate
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