Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.
(1/2)x + (1/3)y = 2
(3/2)x - (1/2)y = -12

Verified step by step guidance

1. Write the system of equations in standard form: \[\frac{1}{2}x + \frac{1}{3}y = 2\] \[\frac{3}{2}x - \frac{1}{2}y = -12\]
2. Identify the coefficients for the variables to form the coefficient matrix: \[A = \begin{bmatrix} \frac{1}{2} & \frac{1}{3} \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}\]
3. Calculate the determinant of matrix \(A\), denoted as \(D\), using the formula: \[D = a_{11}a_{22} - a_{12}a_{21}\] where \(a_{11} = \frac{1}{2}\), \(a_{12} = \frac{1}{3}\), \(a_{21} = \frac{3}{2}\), and \(a_{22} = -\frac{1}{2}\).
4. If \(D \neq 0\), find determinants \(D_x\) and \(D_y\) by replacing the respective columns of \(A\) with the constants vector: \[D_x = \begin{vmatrix} 2 & \frac{1}{3} \\ -12 & -\frac{1}{2} \end{vmatrix}, \quad D_y = \begin{vmatrix} \frac{1}{2} & 2 \\ \frac{3}{2} & -12 \end{vmatrix}\]
5. Solve for \(x\) and \(y\) using Cramer's rule: \[x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}\] If \(D = 0\), use another method such as substitution or elimination to find the solution set.

Key Concepts

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. For a system of two equations, it involves calculating the determinant of the coefficient matrix (D) and determinants of matrices formed by replacing columns with constants. If D ≠ 0, the system has a unique solution found by dividing these determinants by D.

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This value helps determine if the system has a unique solution (nonzero determinant) or if it is dependent or inconsistent (zero determinant). It is essential for applying Cramer's Rule correctly.

Alternative Methods for Solving Systems When D = 0

If the determinant D equals zero, Cramer's Rule cannot be used because the system may have infinitely many solutions or no solution. In such cases, methods like substitution, elimination, or analyzing the system's consistency are used to find the solution set or determine if no solution exists.