Math Problem Statement
Solution
To solve the system of equations using a matrix, we will express the equations in matrix form, solve using the inverse of the coefficient matrix (if possible), and find the solution.
Step 1: Write the system as a matrix equation
The system is:
This can be written as:
1 & 7 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 17 \\ 4 \end{bmatrix}$$ Here: - The coefficient matrix $$A = \begin{bmatrix} 1 & 7 \\ 2 & -1 \end{bmatrix}$$, - The variable matrix $$\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}$$, - The constant matrix $$\mathbf{b} = \begin{bmatrix} 17 \\ 4 \end{bmatrix}$$. ### Step 2: Solve using the formula $$\mathbf{x} = A^{-1} \mathbf{b}$$ To find $$\mathbf{x}$$, we need the inverse of $$A$$. The inverse of a 2x2 matrix $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix},$$ where $$\text{det}(A) = ad - bc$$. #### Step 3: Compute $$\text{det}(A)$$ For $$A = \begin{bmatrix} 1 & 7 \\ 2 & -1 \end{bmatrix}$$: $$\text{det}(A) = (1)(-1) - (2)(7) = -1 - 14 = -15$$ Since $$\text{det}(A) \neq 0$$, the matrix is invertible. #### Step 4: Compute $$A^{-1}$$ $$A^{-1} = \frac{1}{-15} \begin{bmatrix} -1 & -7 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{15} & \frac{7}{15} \\ \frac{2}{15} & -\frac{1}{15} \end{bmatrix}$$ #### Step 5: Multiply $$A^{-1} \mathbf{b}$$ $$\mathbf{x} = A^{-1} \mathbf{b} = \begin{bmatrix} \frac{1}{15} & \frac{7}{15} \\ \frac{2}{15} & -\frac{1}{15} \end{bmatrix} \begin{bmatrix} 17 \\ 4 \end{bmatrix}$$ Perform the matrix multiplication: $$\begin{bmatrix} \frac{1}{15} & \frac{7}{15} \\ \frac{2}{15} & -\frac{1}{15} \end{bmatrix} \begin{bmatrix} 17 \\ 4 \end{bmatrix} = \begin{bmatrix} \frac{1}{15}(17) + \frac{7}{15}(4) \\ \frac{2}{15}(17) - \frac{1}{15}(4) \end{bmatrix} = \begin{bmatrix} \frac{17 + 28}{15} \\ \frac{34 - 4}{15} \end{bmatrix} = \begin{bmatrix} \frac{45}{15} \\ \frac{30}{15} \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$$ ### Final Solution: $$x = 3, \, y = 2$$ Would you like a more detailed explanation or further assistance? --- ### Related Questions: 1. How do you verify the solution by substituting $$x = 3$$ and $$y = 2$$ into the original equations? 2. What happens if the determinant of the coefficient matrix is zero? 3. How can matrix inversion methods be extended to larger systems of equations? 4. What are the alternative methods to solve this system of equations, such as Gaussian elimination? 5. How does the concept of matrix determinants relate to the solvability of systems of equations? ### Tip: Always check the determinant before computing the inverse of a matrix to ensure the system is solvable!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Inversion
Formulas
Matrix equation: A * X = B
Determinant of a 2x2 matrix: det(A) = ad - bc
Inverse of a 2x2 matrix: A^-1 = (1/det(A)) * [[d, -b], [-c, a]]
Theorems
Matrix Inversion Theorem
Suitable Grade Level
Grades 9-12
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