Given the system of equations, match the following items. \( x+2 y=7 \) \( x-2 y=-1 \) 1. \( \left[\begin{array}{cc}1 & 2 \\ 1 & -2\end{array}\right] \) 2. \( \left[\begin{array}{cc}7 & 2 \\ -1 & -2\end{array}\right] \) 3. \( \left[\begin{array}{cc}1 & 7 \\ 1 & -1\end{array}\right] \) system determinant y determinant

To find the determinants for the given system, we can start with the coefficient matrix for the system of equations. The coefficient matrix is represented by \( \left[\begin{array}{cc}1 & 2 \\ 1 & -2\end{array}\right] \). Calculating the determinant for the coefficient matrix: \(\text{Determinant} = (1)(-2) - (2)(1) = -2 - 2 = -4\). Therefore, this corresponds to item 1 as the system determinant. For the y determinant, we replace the second column of the coefficient matrix with the constant terms from the right-hand side of the equations. So we construct the matrix \( \left[\begin{array}{cc}1 & 7 \\ 1 & -1\end{array}\right] \). The determinant is calculated as follows: \(\text{Determinant} = (1)(-1) - (7)(1) = -1 - 7 = -8\). This corresponds to item 3 for the y determinant. So the matches are: - System determinant: Item 1 - Y determinant: Item 3