Math Problem Statement
4 -1 = 10 -2 5 = 4 solve using matrix
Solution
The system of linear equations you have provided is:
This system can be solved using matrices. Let's rewrite the system in matrix form:
4 & -1 \\ -2 & 5 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 10 \\ 4 \end{pmatrix}$$ ### Step 1: Matrix Representation The matrix equation is written as: $$A \mathbf{x} = \mathbf{b}$$ Where: - $$A$$ is the coefficient matrix: $$A = \begin{pmatrix} 4 & -1 \\ -2 & 5 \end{pmatrix}$$ - $$\mathbf{x}$$ is the variable matrix: $$\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$$ - $$\mathbf{b}$$ is the constant matrix: $$\mathbf{b} = \begin{pmatrix} 10 \\ 4 \end{pmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ To solve for $$\mathbf{x}$$, we use the formula: $$\mathbf{x} = A^{-1} \mathbf{b}$$ Let's calculate the inverse of matrix $$A$$. The inverse of a 2x2 matrix is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ For the matrix $$A = \begin{pmatrix} 4 & -1 \\ -2 & 5 \end{pmatrix}$$, the determinant $$\text{det}(A)$$ is: $$\text{det}(A) = (4)(5) - (-1)(-2) = 20 - 2 = 18$$ So, the inverse is: $$A^{-1} = \frac{1}{18} \begin{pmatrix} 5 & 1 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix} \frac{5}{18} & \frac{1}{18} \\ \frac{2}{18} & \frac{4}{18} \end{pmatrix}$$ ### Step 3: Multiply $$A^{-1}$$ by $$\mathbf{b}$$ Now multiply the inverse matrix by the constant matrix: $$A^{-1} \mathbf{b} = \begin{pmatrix} \frac{5}{18} & \frac{1}{18} \\ \frac{2}{18} & \frac{4}{18} \end{pmatrix} \begin{pmatrix} 10 \\ 4 \end{pmatrix}$$ Carrying out the multiplication: $$\mathbf{x} = \begin{pmatrix} \frac{5}{18}(10) + \frac{1}{18}(4) \\ \frac{2}{18}(10) + \frac{4}{18}(4) \end{pmatrix} = \begin{pmatrix} \frac{50}{18} + \frac{4}{18} \\ \frac{20}{18} + \frac{16}{18} \end{pmatrix} = \begin{pmatrix} \frac{54}{18} \\ \frac{36}{18} \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$ ### Final Solution The solution is: $$x = 3, \quad y = 2$$ Would you like more details or have any questions? ### Related Questions: 1. How do you calculate the determinant of a 3x3 matrix? 2. What is the geometric interpretation of solving a system of equations using matrices? 3. How can matrix multiplication be applied in real-world scenarios? 4. How does the Cramer's rule differ from using matrix inverses to solve systems? 5. Can all systems of equations be solved using matrix inverses? ### Tip: When finding the inverse of a 2x2 matrix, always check that the determinant is non-zero; otherwise, the matrix has no inverse, and the system may not have a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Inversion
Formulas
Matrix Equation: A * x = b
Inverse of 2x2 Matrix: A^(-1) = (1/det(A)) * adjugate(A)
Determinant of 2x2 Matrix: det(A) = ad - bc
Theorems
Matrix Inversion
Determinant Theorem
Suitable Grade Level
Grades 10-12
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