Show Dramples Qaestion Solve the following system of equations using an inverse matrix. You must also indicate the inverse matrix, A^(-1), that was used to solve the system. You may optionally write the inverse matrix with a scalar coefficient. {:[-2x+7y=9],[-3x+8y=4]:} Answer antenticeat of A^(-1)=◻[◻quad x=◻quad y=◻ shang

Question

Show Dramples Qaestion Solve the following system of equations using an inverse matrix. You must also indicate the inverse matrix,  A^(-1), that was used to solve the system. You may optionally write the inverse matrix with a scalar coefficient.  {:[-2x+7y=9],[-3x+8y=4]:} Answer antenticeat of  A^(-1)=◻[◻quad x=◻quad y=◻ shang

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Write Equations in Matrix Form:  First, let's write down the system of equations in matrix form. We have: $$ \begin{bmatrix} -2 & 7 \ -3 & 8 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 9 \ 4 \end{bmatrix} $$Calculate Determinant of Matrix A:  Next, calculate the determinant of the matrix A, where A is: $$ \begin{bmatrix} -2 & 7 \ -3 & 8 \end{bmatrix} $$ The determinant, det(A), is calculated as: $$ (-2)(8) - (7)(-3) = -16 + 21 = 5 $$Find Inverse of Matrix A:  Now, find the inverse of matrix A, A^(-1). The formula for the inverse of a 2x2 matrix is: $$ \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} $$ Applying this to our matrix: $$ A^{-1} = \frac{1}{5} \begin{bmatrix} 8 & -7 \ 3 & -2 \end{bmatrix} $$Multiply Inverse Matrix by Constant Matrix:  Multiply the inverse matrix A^(-1) by the constant matrix on the right side of the equation to find the values of x and y: $$ \begin{bmatrix} 8 & -7 \ 3 & -2 \end{bmatrix} \begin{bmatrix} 9 \ 4 \end{bmatrix} = \begin{bmatrix} 8*9 + (-7)*4 \ 3*9 + (-2)*4 \end{bmatrix} = \begin{bmatrix} 72 - 28 \ 27 - 8 \end{bmatrix} = \begin{bmatrix} 44 \ 19 \end{bmatrix} $$ However, we forgot to divide by the determinant, which is 5. Correcting this: $$ x = \frac{44}{5}, \quad y = \frac{19}{5} $$

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