For Exercises 11–22, use Cramer's Rule to solve each system. 2x = 3y + 25x = 51 - 4y

Verified step by step guidance

1. Rewrite the system of equations in standard form: 2x - 3y = 2 and 5x + 4y = 51.
2. Identify the coefficient matrix A, the variable matrix X, and the constant matrix B.
3. Calculate the determinant of matrix A, det(A).
4. Find the determinant of the matrix formed by replacing the first column of A with B, det(A_x).
5. Find the determinant of the matrix formed by replacing the second column of A with B, det(A_y).

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 determinant of the coefficient matrix is non-zero, the system has a unique solution, which can be found using the determinants of modified matrices formed by replacing one column of the coefficient matrix with the constants from the equations.

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties about 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 determining the solutions of the 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 solving it involves finding the intersection point of the lines represented by these equations.