For Exercises 11–22, use Cramer's Rule to solve each system. x + 2y = 33x - 4y = 4

Verified step by step guidance

1. Identify the coefficient matrix A from the system of equations: A = \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix}.
2. Calculate the determinant of matrix A, denoted as det(A).
3. Form the matrix A_x by replacing the first column of A with the constants from the right side of the equations: A_x = \begin{bmatrix} 3 & 2 \\ 4 & -4 \end{bmatrix}.
4. Calculate the determinant of matrix A_x, denoted as det(A_x).
5. Use Cramer's Rule to find x: x = \frac{\text{det}(A_x)}{\text{det}(A)}.

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 by calculating 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 existence and uniqueness of solutions to 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 for the variables that satisfy all equations simultaneously. Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines), and methods like substitution, elimination, and Cramer's Rule can be used to find these solutions.