Solve for \( Y \) in the matrix equation below: \[ -9\left[\begin{array}{ccc} x+4 & 8 & -3 \\ 1 & 22 & 2 y \\ 7 & -2 & z+2 \end{array}\right]=\left[\begin{array}{ccc} -18 x-81 & -72 & 27 \\ -9 & -198 & 72 \\ -63 & 18 & -99 \end{array}\right. \] \( y=-4 \) \( y=-5 \) \( y=10 \) \( y=9 \)

To solve for \( Y \) in the matrix equation, let's start by simplifying the left side. Multiplying the matrix by \(-9\) means we distribute \(-9\) to each entry in the matrix: \[ \left[\begin{array}{ccc} -9(x+4) & -72 & 27 \\ -9 & -198 & -18y \\ -63 & 18 & -9(z+2) \end{array}\right] \] Now, let's compare the two matrices entry by entry. 1. From the second entry of the second row: \[ -18y = 72 \rightarrow y = -\frac{72}{18} = -4 \] Now, let's compare the possible values of \( y \) given: - \( y = -4 \) - \( y = -5 \) - \( y = 10 \) - \( y = 9 \) The solution we found, \( y = -4 \), is one of the options provided. Therefore, the correct answer is \( \boxed{-4} \). It's essential to verify this solution by checking other entries as well, but we can be confident as we found a valid \( y \).