For Exercises 11–22, use Cramer's Rule to solve each system. 4x - 5y = 172x + 3y = 3

Verified step by step guidance

1. Identify the coefficients from the system of equations: 4x - 5y = 17 and 2x + 3y = 3.
2. Construct the coefficient matrix A: \( A = \begin{bmatrix} 4 & -5 \\ 2 & 3 \end{bmatrix} \).
3. Calculate the determinant of matrix A, denoted as det(A): \( \text{det}(A) = (4)(3) - (-5)(2) \).
4. Construct the matrix A_x by replacing the first column of A with the constants from the equations: \( A_x = \begin{bmatrix} 17 & -5 \\ 3 & 3 \end{bmatrix} \).
5. Calculate the determinant of matrix A_x, denoted as det(A_x): \( \text{det}(A_x) = (17)(3) - (-5)(3) \).

Key Concepts

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, using determinants. It states that if the system can be represented in matrix form, the solution for each variable can be found by taking the ratio of the determinant of a modified matrix to the determinant of the coefficient matrix.

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties of the matrix, such as whether it is invertible. For a 2x2 matrix, the determinant is calculated as ad - bc, where the matrix is represented as [[a, b], [c, d]]. Determinants are crucial in Cramer's Rule for finding the values of variables in a system of equations.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The solution to the system is the set of values that satisfy all equations simultaneously. In this case, the system includes two equations in two variables (x and y), and the goal is to find the intersection point of their graphs, which represents the solution.