Math Problem Statement

Solve by the method of matrix inversion: 5x + 6y = 39 7x - 8y = -11

Solution

We are asked to solve the system of equations using the method of matrix inversion. The system is:

5x + 6y &= 39 \\ 7x - 8y &= -11 \end{aligned}$$ This can be written in matrix form as: $$\begin{pmatrix} 5 & 6 \\ 7 & -8 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 39 \\ -11 \end{pmatrix}$$ Let $$A = \begin{pmatrix} 5 & 6 \\ 7 & -8 \end{pmatrix}$$, and the solution vector is $$\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$$, and the constant vector $$\mathbf{b} = \begin{pmatrix} 39 \\ -11 \end{pmatrix}$$. To solve for $$\mathbf{x}$$, we use the formula: $$\mathbf{x} = A^{-1} \mathbf{b}$$ ### Step 1: Find the inverse of matrix $$A$$ The inverse of a 2x2 matrix $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ where the determinant $$\text{det}(A)$$ is: $$\text{det}(A) = ad - bc$$ For our matrix $$A = \begin{pmatrix} 5 & 6 \\ 7 & -8 \end{pmatrix}$$, the determinant is: $$\text{det}(A) = (5)(-8) - (6)(7) = -40 - 42 = -82$$ Thus, the inverse of $$A$$ is: $$A^{-1} = \frac{1}{-82} \begin{pmatrix} -8 & -6 \\ -7 & 5 \end{pmatrix}$$ $$A^{-1} = \begin{pmatrix} \frac{8}{82} & \frac{6}{82} \\ \frac{7}{82} & \frac{-5}{82} \end{pmatrix}$$ $$A^{-1} = \begin{pmatrix} \frac{4}{41} & \frac{3}{41} \\ \frac{7}{82} & \frac{-5}{82} \end{pmatrix}$$ ### Step 2: Multiply $$A^{-1}$$ by $$\mathbf{b}$$ Now, multiply the inverse matrix $$A^{-1}$$ by the constant vector $$\mathbf{b}$$: $$\mathbf{x} = \begin{pmatrix} \frac{4}{41} & \frac{3}{41} \\ \frac{7}{82} & \frac{-5}{82} \end{pmatrix} \begin{pmatrix} 39 \\ -11 \end{pmatrix}$$ Performing the matrix multiplication: $$x = \frac{4}{41} \times 39 + \frac{3}{41} \times (-11) = \frac{156}{41} + \frac{-33}{41} = \frac{123}{41} = 3$$ $$y = \frac{7}{82} \times 39 + \frac{-5}{82} \times (-11) = \frac{273}{82} + \frac{55}{82} = \frac{328}{82} = 4$$ ### Final Solution The solution is: $$x = 3, \quad y = 4$$ Would you like more details or have any questions? --- Here are some related questions: 1. What is the determinant of a matrix, and how is it used in finding the inverse? 2. How can you verify the solution of a system of equations using matrix multiplication? 3. Can matrix inversion be applied to larger systems of equations? 4. What are the conditions for a matrix to be invertible? 5. What are some alternative methods for solving systems of equations (e.g., Cramer's rule)? **Tip:** When dealing with matrices, always check if the determinant is zero before attempting to find the inverse—if it is, the matrix is non-invertible.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Systems of Linear Equations

Formulas

Inverse of a 2x2 matrix formula: A^{-1} = 1/det(A) * [[d, -b], [-c, a]]
Determinant of a 2x2 matrix: det(A) = ad - bc
Matrix multiplication

Theorems

Matrix inversion theorem

Suitable Grade Level

Grades 10-12

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