Math Problem Statement

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Consider the system: Given that b ≠ -1, for which value of the parameter a does the system have no solutions? This system has a unique solution for all except one value of a. What is this value of a? For which values of a, b does this system have infinitely many solutions?

Solution

The system of equations presented appears to be:

3 & -1 & | & 2 \\ -3 & 3 + 8a & | & -6 - 4b \end{pmatrix}$$ The question asks for: ### Part a: Given $$b \neq -1$$, for which value of $$a$$ does the system have no solutions? ### Part b: For which value of $$a$$ does the system have a unique solution for all except one value of $$a$$? ### Part c: For which values of $$a$$ and $$b$$ does this system have infinitely many solutions? --- Let's approach each part by analyzing the augmented matrix and checking conditions for no solutions, unique solutions, and infinitely many solutions based on the determinant and row reductions. I'll solve this step by step.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Augmented Matrices

Formulas

Determinants of matrices
Row reduction methods for solving systems of linear equations

Theorems

Condition for no solutions (inconsistent system)
Condition for unique solution (determinant ≠ 0)
Condition for infinitely many solutions (dependent system)

Suitable Grade Level

Undergraduate Mathematics or Advanced High School

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