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. x + y = 4 2x - y = 2

Verified step by step guidance

1. First, write the system of equations in matrix form: \( \begin{bmatrix} 1 & 1 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ 2 \end{bmatrix} \).
2. Calculate the determinant \( D \) of the coefficient matrix \( \begin{bmatrix} 1 & 1 \\ 2 & -1 \end{bmatrix} \) using the formula \( D = ad - bc \), where \( a = 1, b = 1, c = 2, d = -1 \).
3. If \( D \neq 0 \), use Cramer's Rule to find \( x \) and \( y \). Calculate \( D_x \) by replacing the first column of the coefficient matrix with the constants from the right-hand side, and \( D_y \) by replacing the second column.
4. Calculate \( x = \frac{D_x}{D} \) and \( y = \frac{D_y}{D} \).
5. If \( D = 0 \), use another method such as substitution or elimination to solve the system of equations.

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 each variable by substituting the constants into the determinant formula.

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. In the context of Cramer's Rule, the determinant of the coefficient matrix (D) indicates whether the system has a unique solution (D ≠ 0) or if it is either inconsistent or has infinitely many solutions (D = 0).

Alternative Methods for D = 0

When the determinant of the coefficient matrix is zero (D = 0), it indicates that the system of equations may be dependent or inconsistent. In such cases, alternative methods like substitution or elimination can be used to analyze the system further, helping to determine if there are infinitely many solutions or no solution at all.