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. $\newline$ $\begin{cases} -2x-5y=-6 \ -x+y=8 \end{cases}$ $\newline$ $A^{-1}=\square[\square]\quad x=\square\quad y=\square$

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Q. 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.

\newline

\begin{cases} -2x-5y=-6 \ -x+y=8 \end{cases}

\newline

A^{-1}=\square[\square]\quad x=\square\quad y=\square

Write System of Equations: Step $1$ : Write the system of equations in matrix form. $\newline$ We have the system: $\newline$ $-2x - 5y = -6$ $\newline$ $-x + y = 8$ $\newline$ This can be represented as: $\newline$ $A * [x, y] = [b]$ $\newline$ Where $A = [[-2, -5], [-1, 1]]$ and $b = [-6, 8]$ .
Calculate Determinant of Matrix: Step $2$ : Calculate the determinant of matrix $A$ . The determinant of $A$ , $\text{det}(A)$ , is calculated as: $\text{det}(A) = (-2)(1) - (-5)(-1) = -2 - 5 = -7$ .
Find Inverse of Matrix: Step $3$ : Find the inverse of matrix $A$ , $A^{-1}$ . $\newline$ Using the formula for the inverse of a $2$ x $2$ matrix, $A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A)$ . $\newline$ $\text{adj}(A)$ is the adjugate of $A$ , which is $\begin{bmatrix} 1 & 5 \ 1 & -2 \end{bmatrix}$ . $\newline$ So, $A^{-1} = \frac{1}{-7} \times \begin{bmatrix} 1 & 5 \ 1 & -2 \end{bmatrix} = \begin{bmatrix} -\frac{1}{7} & -\frac{5}{7} \ -\frac{1}{7} & \frac{2}{7} \end{bmatrix}$ .
Multiply Inverse Matrix: Step $4$ : Multiply the inverse matrix $A^{-1}$ by the matrix $b$ to find the values of $x$ and $y$ . $\newline$ Using matrix multiplication, we get: $\newline$ A^{$-1$} \cdot b = \left[\begin{array}{cc}$\newline$-\frac{$1$}{$7$} & -\frac{$5$}{$7$} (\newline\)-\frac{$1$}{$7$} & \frac{$2$}{$7$}$\newline$\end{array}\right] \cdot \left[\begin{array}{c}$\newline$$-6$ (\newline\)$8$$\newline$\end{array}\right] $\newline$ $= \left[\left(-\frac{1}{7}\right)(-6) + \left(-\frac{5}{7}\right)(8), \left(-\frac{1}{7}\right)(-6) + \left(\frac{2}{7}\right)(8)\right]$ $\newline$ $= \left[\frac{6}{7} - \frac{40}{7}, \frac{6}{7} + \frac{16}{7}\right]$ $\newline$ $= \left[-\frac{34}{7}, \frac{22}{7}\right]$ $\newline$ $= [$-4$.$857$, $3$.$143$].