Math Problem Statement

solve the system using matrix -2x-3y=-26 3x+4y=36

Solution

We can solve the system of equations using matrices.

The system is:

-2x - 3y &= -26 \\ 3x + 4y &= 36 \end{aligned}$$ We can express this system in matrix form as: $$\begin{bmatrix} -2 & -3 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -26 \\ 36 \end{bmatrix}$$ This is in the form $$A \cdot X = B$$, where: $$A = \begin{bmatrix} -2 & -3 \\ 3 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} -26 \\ 36 \end{bmatrix}$$ ### Step 1: Find the inverse of matrix $$A$$. The inverse of a 2x2 matrix $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ is: $$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ For our matrix $$A = \begin{bmatrix} -2 & -3 \\ 3 & 4 \end{bmatrix}$$, we calculate the determinant: $$\text{Det}(A) = (-2)(4) - (-3)(3) = -8 + 9 = 1$$ Thus, the inverse of $$A$$ is: $$A^{-1} = \frac{1}{1} \begin{bmatrix} 4 & 3 \\ -3 & -2 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ -3 & -2 \end{bmatrix}$$ ### Step 2: Multiply the inverse of $$A$$ by $$B$$ to find $$X$$. Now, we multiply $$A^{-1}$$ by $$B$$: $$X = A^{-1} \cdot B = \begin{bmatrix} 4 & 3 \\ -3 & -2 \end{bmatrix} \begin{bmatrix} -26 \\ 36 \end{bmatrix}$$ Performing the multiplication: $$\begin{aligned} x &= (4)(-26) + (3)(36) = -104 + 108 = 4 \\ y &= (-3)(-26) + (-2)(36) = 78 - 72 = 6 \end{aligned}$$ ### Final Answer: The solution to the system is: $$x = 4, \quad y = 6$$ Would you like further explanation or have any questions? ### Related Questions: 1. How do you solve a system of three equations using matrices? 2. How do you calculate the determinant of a 3x3 matrix? 3. What happens if the determinant of the matrix is zero? 4. How do you perform matrix multiplication step by step? 5. How would the solution change if the system had no solution or infinitely many solutions? **Tip:** Always check your final solution by substituting the values of $$x$$ and $$y$$ back into the original equations to verify the correctness.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Systems of Equations

Formulas

Matrix equation: A * X = B
Inverse of a 2x2 matrix: A^-1 = (1/det(A)) * [d -b; -c a]
Determinant of a 2x2 matrix: det(A) = ad - bc
Matrix multiplication rules

Theorems

Inverse Matrix Theorem
Determinant Theorem

Suitable Grade Level

Grades 9-12

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