Solve System of Linear Equations Using Matrix Method

Math Problem Statement

Solve the system of linear equations by matrix method: 2x + 3y - z = 5, 4x + 4y - 3z = 2, 2x - 3y + 2z = 2.

Solution

The system of linear equations provided in the image is:

1. & \quad 2x + 3y - z = 5 \\ 2. & \quad 4x + 4y - 3z = 2 \\ 3. & \quad 2x - 3y + 2z = 2 \end{aligned}$$ We will solve this using the matrix method, i.e., representing the system in matrix form $$AX = B$$, where: - $$A$$ is the coefficient matrix, - $$X$$ is the variable matrix, - $$B$$ is the constant matrix. The matrix form for the above system is: $$A = \begin{pmatrix} 2 & 3 & -1 \\ 4 & 4 & -3 \\ 2 & -3 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 5 \\ 2 \\ 2 \end{pmatrix}.$$ Now, we will solve this system by finding $$X = A^{-1}B$$, which requires calculating the inverse of matrix $$A$$ and then multiplying it by matrix $$B$$. Let me compute this for you.The solution to the system of equations is: $$x = 0.875, \quad y = 2.25, \quad z = 3.5$$ This is the solution using the matrix method. Would you like further details on the steps involved, or have any questions? Here are five related questions that could deepen your understanding: 1. How is the inverse of a matrix calculated? 2. What are the conditions for a matrix to have an inverse? 3. Can this system be solved using another method like Gaussian elimination? 4. What are some practical applications of solving systems of equations with matrices? 5. How do we verify the solution to a system of linear equations? **Tip:** When using the matrix method, it's important to check if the determinant of the matrix is non-zero, as a zero determinant means the matrix is not invertible.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Inversion

Formulas

Matrix equation: AX = B
Inverse of a matrix: A^(-1)

Theorems

Inverse Matrix Theorem
Cramer's Rule

Suitable Grade Level

Grades 10-12 (High School Level)

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