For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}4x - 5y = 17 \\2x + 3y = 3\end{cases}$

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1. Write the system of equations in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A = \begin{bmatrix} 4 & -5 \\ 2 & 3 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}\), and \(\mathbf{b} = \begin{bmatrix} 17 \\ 3 \end{bmatrix}\).
2. Calculate the determinant of matrix \(A\), denoted as \(\det(A)\), using the formula \(\det(A) = a_{11}a_{22} - a_{12}a_{21}\). For this matrix, compute \(\det(A) = 4 \times 3 - (-5) \times 2\).
3. Form matrix \(A_x\) by replacing the first column of \(A\) with vector \(\mathbf{b}\), so \(A_x = \begin{bmatrix} 17 & -5 \\ 3 & 3 \end{bmatrix}\). Then calculate \(\det(A_x)\) using the determinant formula.
4. Form matrix \(A_y\) by replacing the second column of \(A\) with vector \(\mathbf{b}\), so \(A_y = \begin{bmatrix} 4 & 17 \\ 2 & 3 \end{bmatrix}\). Then calculate \(\det(A_y)\) using the determinant formula.
5. Use Cramer's Rule to find the solutions: \(x = \frac{\det(A_x)}{\det(A)}\) and \(y = \frac{\det(A_y)}{\det(A)}\). These fractions give the values of \(x\) and \(y\) that solve the system.

Key Concepts

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the system has the same number of equations as unknowns and the determinant of the coefficient matrix is non-zero. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.

Determinants of 2x2 Matrices

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This value helps determine if the system has a unique solution (non-zero determinant) or not. Determinants are essential in Cramer's Rule to find the values of variables by comparing determinants of modified matrices.

Solving Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and matrix approaches like Cramer's Rule, which is efficient for small systems.