Converting a System of Linear Equations to a Matrix Equation

Math Problem Statement

3x-5y+2z=-8 7x+5y-3z=16 5x+3y-7z=0 to matrix equation

To convert the given system of equations into a matrix equation, follow these steps:

3x - 5y + 2z &= -8 \\ 7x + 5y - 3z &= 16 \\ 5x + 3y - 7z &= 0 \end{aligned}$$ 2. Express this system as a matrix equation of the form **Ax = b**, where: - **A** is the matrix of coefficients, - **x** is the vector of variables $$[x, y, z]^T$$, - **b** is the vector of constants. The matrix form of the system is: $$\begin{pmatrix} 3 & -5 & 2 \\ 7 & 5 & -3 \\ 5 & 3 & -7 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -8 \\ 16 \\ 0 \end{pmatrix}$$ This is the matrix equation representation of the given system of equations. Would you like details on how to solve this system or any further explanations? Here are 5 follow-up questions: 1. How can you solve a system of linear equations using matrix inversion? 2. What is the determinant of a 3x3 matrix, and how does it affect solving the system? 3. How do you solve a system using Cramer's Rule? 4. What are the steps to solve a system of equations using Gaussian elimination? 5. What role does matrix rank play in determining if a system has a unique solution? **Tip:** When solving systems of equations, always check if the determinant of the coefficient matrix is non-zero. If it's zero, the system may not have a unique solution.

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Linear Algebra
Matrix Representation
Systems of Linear Equations

Matrix Equation: Ax = b
Matrix Inversion
Determinant of a 3x3 Matrix

Cramer's Rule
Gaussian Elimination
Matrix Rank Theorem

Grades 10-12

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