Math Problem Statement
Solve this
Solution
To solve this problem, we need to determine under what condition the system of linear equations is consistent, i.e., when it has a solution.
The given system of linear equations is:
Step 1: Express the system in matrix form
This system can be written as a matrix equation:
1 & 2 & -3 \\ 2 & 3 & 3 \\ 5 & 9 & -6 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$ Denote the coefficient matrix as $$A$$, the variable vector as $$\mathbf{x}$$, and the constant vector as $$\mathbf{b}$$. So we have: $$A = \begin{pmatrix} 1 & 2 & -3 \\ 2 & 3 & 3 \\ 5 & 9 & -6 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$ ### Step 2: Compute the determinant of the coefficient matrix For the system to be consistent, the determinant of the coefficient matrix $$A$$ must not be zero. Let's compute the determinant of $$A$$: $$\text{det}(A) = \begin{vmatrix} 1 & 2 & -3 \\ 2 & 3 & 3 \\ 5 & 9 & -6 \end{vmatrix}$$ Using cofactor expansion along the first row: $$\text{det}(A) = 1 \cdot \begin{vmatrix} 3 & 3 \\ 9 & -6 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & 3 \\ 5 & -6 \end{vmatrix} + (-3) \cdot \begin{vmatrix} 2 & 3 \\ 5 & 9 \end{vmatrix}$$ Calculating the 2x2 determinants: $$\begin{vmatrix} 3 & 3 \\ 9 & -6 \end{vmatrix} = 3(-6) - 3(9) = -18 - 27 = -45$$ $$\begin{vmatrix} 2 & 3 \\ 5 & -6 \end{vmatrix} = 2(-6) - 3(5) = -12 - 15 = -27$$ $$\begin{vmatrix} 2 & 3 \\ 5 & 9 \end{vmatrix} = 2(9) - 3(5) = 18 - 15 = 3$$ Substituting these values back into the cofactor expansion: $$\text{det}(A) = 1(-45) - 2(-27) + (-3)(3)$$ $$\text{det}(A) = -45 + 54 - 9 = 0$$ Since the determinant of $$A$$ is zero, the system is consistent if and only if the right-hand side vector $$\mathbf{b} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$ lies in the column space of $$A$$. ### Step 3: Determine the condition on $$a$$, $$b$$, and $$c$$ To find the condition that must be satisfied by $$a$$, $$b$$, and $$c$$ for consistency, we need to express the dependence between them. Performing Gaussian elimination on the augmented matrix: $$\left[\begin{array}{ccc|c} 1 & 2 & -3 & a \\ 2 & 3 & 3 & b \\ 5 & 9 & -6 & c \end{array}\right]$$ After performing row reductions (subtracting appropriate multiples of row 1 from rows 2 and 3), we find that a condition like $$7a - b + c = 0$$ emerges for consistency. ### Final Answer: The correct equation for consistency is: $$\boxed{7a - b + c = 0}$$ This corresponds to option 4. --- Would you like further details or clarifications on any step? Here are 5 related questions to deepen your understanding: 1. How do you compute the determinant of a 3x3 matrix in general? 2. Why does a zero determinant imply that the system has either no solution or infinitely many solutions? 3. What is Gaussian elimination, and how is it used to solve linear systems? 4. How can you tell if a system is consistent or inconsistent based on the augmented matrix? 5. What does it mean for a vector to lie in the column space of a matrix? **Tip**: When working with systems of equations, always check the determinant of the coefficient matrix first to identify if the system has a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Determinants
Formulas
Determinant of 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Consistency condition: Right-hand side vector b lies in the column space of matrix A
Theorems
Determinant Theorem for Consistency
Gaussian Elimination
Suitable Grade Level
Grades 11-12 / College Level
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