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. 1.5
x + 3y = 5
2x + 4y = 3

Verified step by step guidance

1. Write the system of equations in standard form: \$1.5x + 3y = 5$ \$2x + 4y = 3$.
2. Identify the coefficients to form the coefficient matrix \(A\): \(A = \begin{bmatrix} 1.5 & 3 \\ 2 & 4 \end{bmatrix}\).
3. Calculate the determinant \(D\) of matrix \(A\) using the formula: \(D = (1.5)(4) - (3)(2)\).
4. If \(D \neq 0\), find the determinants \(D_x\) and \(D_y\) by replacing the respective columns of \(A\) with the constants vector \(\begin{bmatrix} 5 \\ 3 \end{bmatrix}\): \(D_x = \begin{vmatrix} 5 & 3 \\ 3 & 4 \end{vmatrix}\), \(D_y = \begin{vmatrix} 1.5 & 5 \\ 2 & 3 \end{vmatrix}\).
5. Use Cramer's rule to solve for \(x\) and \(y\): \(x = \frac{D_x}{D}\), \(y = \frac{D_y}{D}\). If \(D = 0\), then 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. It applies when the system has the same number of equations as unknowns and the determinant of the coefficient matrix (D) is nonzero. The solution for each variable is found by replacing the corresponding column in the coefficient matrix with the constants from the equations and calculating the determinant.

Determinant of a Matrix

The determinant is a scalar value that can be computed from a square matrix and provides important information about the matrix, such as invertibility. For a 2x2 matrix, the determinant is calculated as ad - bc. If the determinant is zero, the matrix is singular, meaning the system may have no solution or infinitely many solutions.

Alternative Methods for Solving Systems

When the determinant D equals zero, Cramer's Rule cannot be used. Alternative methods include substitution, elimination, or using matrix row operations to find the solution set. These methods help determine whether the system has no solution, one solution, or infinitely many solutions.