Math Problem Statement

Write a matrix to represent the system of linear equations: -9x + 5y = 4 and 9x - 2y = 2.

Solution

The system of linear equations provided in the image is:

-9x + 5y &= 4 \\ 9x - 2y &= 2 \end{aligned}$$ To represent this system in matrix form, we write it as $$A\mathbf{x} = \mathbf{b}$$, where: - $$A$$ is the coefficient matrix (containing the coefficients of $$x$$ and $$y$$), - $$\mathbf{x}$$ is the column vector of variables $$[x \ y]^T$$, - $$\mathbf{b}$$ is the column vector on the right-hand side of the equations. ### Step 1: Coefficient matrix From the system, the coefficients of $$x$$ and $$y$$ in the first equation are $$-9$$ and $$5$$, respectively. In the second equation, the coefficients are $$9$$ and $$-2$$. Thus, the coefficient matrix $$A$$ is: $$A = \begin{bmatrix} -9 & 5 \\ 9 & -2 \end{bmatrix}$$ ### Step 2: Variable matrix The variable matrix $$\mathbf{x}$$ is: $$\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}$$ ### Step 3: Right-hand side matrix The right-hand side matrix $$\mathbf{b}$$ contains the constants from the right-hand side of each equation, which are $$4$$ and $$2$$: $$\mathbf{b} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}$$ ### Full matrix form The system can now be written as: $$\begin{bmatrix} -9 & 5 \\ 9 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}$$ This represents the matrix form of the given system of equations. Would you like further details or have any questions? Here are 5 related questions you might consider: 1. How can you solve this system of linear equations using matrices? 2. What is the determinant of the coefficient matrix? 3. Can this system of equations be solved using the inverse of the matrix? 4. How does this matrix equation relate to Gaussian elimination? 5. What other methods can you use to solve a system of linear equations? **Tip:** When converting systems of equations into matrix form, always ensure that the variables are aligned in the same order in both equations before extracting the coefficients.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrices
Systems of Linear Equations

Formulas

Matrix representation of a system of equations: A * X = B
Coefficient Matrix: A = [[-9, 5], [9, -2]]
Variable Matrix: X = [[x], [y]]
Constant Matrix: B = [[4], [2]]

Theorems

Matrix Equation

Suitable Grade Level

Grades 10-12

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