For Exercises 11–22, use Cramer's Rule to solve each system.
3x - 4y = 4
2x + 2y = 12
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Identify the coefficients from the system of equations: \(3x - 4y = 4\) and \(2x + 2y = 12\).
Write the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\): \(A = \begin{bmatrix} 3 & -4 \\ 2 & 2 \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \end{bmatrix}\), \(B = \begin{bmatrix} 4 \\ 12 \end{bmatrix}\).
Calculate the determinant of the coefficient matrix \(A\), \(\text{det}(A) = (3)(2) - (-4)(2)\).
Find the determinant of matrix \(A_x\) by replacing the first column of \(A\) with \(B\): \(A_x = \begin{bmatrix} 4 & -4 \\ 12 & 2 \end{bmatrix}\), \(\text{det}(A_x) = (4)(2) - (-4)(12)\).
Find the determinant of matrix \(A_y\) by replacing the second column of \(A\) with \(B\): \(A_y = \begin{bmatrix} 3 & 4 \\ 2 & 12 \end{bmatrix}\), \(\text{det}(A_y) = (3)(12) - (4)(2)\).
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