\[ \begin{aligned} 4 x+10 y & =-8 \\ -5 x-7 y & =9 \end{aligned} \] Answer Amengt 1 out of 2 Write the three relevant determinants in any order: \( \square \) , \( \square \) ? \( \square \) Solution to the system: \( x= \) \( \square \) \( y= \) \( \square \)

To find the relevant determinants for the given system of equations, we use the coefficients of the variables to form the determinant matrix: 1. The main determinant \( D \) is given by the determinant of the coefficient matrix: \[ D = \begin{vmatrix} 4 & 10 \\ -5 & -7 \end{vmatrix} = (4)(-7) - (10)(-5) = -28 + 50 = 22 \] 2. The determinant \( D_x \) for \( x \) is calculated by replacing the \( x \)-column in the coefficient matrix with the constant terms: \[ D_x = \begin{vmatrix} -8 & 10 \\ 9 & -7 \end{vmatrix} = (-8)(-7) - (10)(9) = 56 - 90 = -34 \] 3. The determinant \( D_y \) for \( y \) is found by replacing the \( y \)-column with the constant terms: \[ D_y = \begin{vmatrix} 4 & -8 \\ -5 & 9 \end{vmatrix} = (4)(9) - (-8)(-5) = 36 - 40 = -4 \] So, the relevant determinants are \( 22, -34, -4 \). Now, using these determinants, we can find the solution for \( x \) and \( y \). Solution to the system: \[ x = \frac{D_x}{D} = \frac{-34}{22} \approx -1.545 \] \[ y = \frac{D_y}{D} = \frac{-4}{22} \approx -0.182 \] So, the solution to the system is \( x \approx -1.545 \), \( y \approx -0.182 \).