Math Problem Statement

determine a and b for which the system has a unique solution, no solution, infinite solution 3x -2 y + z = b, -5x - 8y + 9z = 3, 2x + y + az = -1

Solution

We have a system of three linear equations with three unknowns xx, yy, and zz:

1. \quad 3x - 2y + z &= b \\ 2. \quad -5x - 8y + 9z &= 3 \\ 3. \quad 2x + y + az &= -1 \end{aligned}$$ To analyze when the system has a unique solution, no solution, or infinitely many solutions, we use the concept of the determinant of the coefficient matrix. The system will have: - **A unique solution** if the determinant of the coefficient matrix is non-zero. - **No solution** if the system is inconsistent (e.g., rows lead to contradictory equations). - **Infinite solutions** if the determinant of the coefficient matrix is zero but the system is consistent. ### Step 1: Coefficient Matrix The coefficient matrix of this system is: $$A = \begin{pmatrix} 3 & -2 & 1 \\ -5 & -8 & 9 \\ 2 & 1 & a \end{pmatrix}$$ ### Step 2: Determinant of the Coefficient Matrix We calculate the determinant of $$A$$. For a $$3 \times 3$$ matrix: \[ \det(A) = 3 \begin{vmatrix} -8 & 9 \\ 1 & a \end{vmatrix} - (-2) \begin{vmatrix} -5 & 9 \\ 2 & a \end{vmatrix} + 1 \begin{vmatrix} -5 &

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Determinants

Formulas

Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

Theorems

Determinant Theorem: A system of linear equations has a unique solution if and only if the determinant of its coefficient matrix is non-zero.
Consistency of Linear Systems: If the determinant is zero, check for consistency to determine if there are no solutions or infinitely many solutions.

Suitable Grade Level

Grades 10-12

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