Use Cramer's rule to solve each system of equations. If D = 0, then use another methodto determine the solution set. See Examples 5–7. 3x + 2y = 4 6x + 4y = 8

Verified step by step guidance

1. Identify the system of equations: \(3x + 2y = 4\) and \(6x + 4y = 8\).
2. Write the system in matrix form: \(AX = B\), where \(A = \begin{bmatrix} 3 & 2 \\ 6 & 4 \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \end{bmatrix}\), and \(B = \begin{bmatrix} 4 \\ 8 \end{bmatrix}\).
3. Calculate the determinant \(D\) of matrix \(A\): \(D = \begin{vmatrix} 3 & 2 \\ 6 & 4 \end{vmatrix} = (3)(4) - (2)(6)\).
4. If \(D = 0\), the system may be dependent or inconsistent. Use another method, such as substitution or elimination, to determine the solution set.
5. If \(D \neq 0\), use Cramer's Rule to find \(x\) and \(y\) by calculating \(D_x\) and \(D_y\) and then \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\).

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, provided the determinant of the coefficient matrix is non-zero. It expresses the solution in terms of determinants, allowing for a straightforward calculation of variable values. If the determinant is zero, the system may have no solution or infinitely many solutions, necessitating alternative methods.

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible. For a 2x2 matrix, the determinant is calculated as ad - bc for a matrix [[a, b], [c, d]]. If the determinant is zero, it indicates that the system of equations represented by the matrix is either dependent or inconsistent.

Alternative Methods for Solving Systems

When Cramer's Rule is not applicable due to a zero determinant, alternative methods such as substitution, elimination, or matrix row reduction can be employed to find the solution set. These methods involve manipulating the equations to isolate variables or simplify the system, allowing for the identification of solutions even when the equations are dependent or inconsistent.