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

Verified step by step guidance

1. Write the system of equations in standard form: \\ \(x + y = 4\) and \$2x - y = 2$.
2. Identify the coefficient matrix and calculate the determinant \(D\): \\ \(D = \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} = (1)(-1) - (1)(2)\).
3. Calculate the determinant \(D_x\) by replacing the first column of the coefficient matrix with the constants from the right side of the equations: \\ \(D_x = \begin{vmatrix} 4 & 1 \\ 2 & -1 \end{vmatrix} = (4)(-1) - (1)(2)\).
4. Calculate the determinant \(D_y\) by replacing the second column of the coefficient matrix with the constants: \\ \(D_y = \begin{vmatrix} 1 & 4 \\ 2 & 2 \end{vmatrix} = (1)(2) - (4)(2)\).
5. Use Cramer's rule to find the solutions for \(x\) and \(y\): \\ \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\). 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 vector and calculating the determinant ratio.

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that indicates whether the matrix is invertible. For a 2x2 matrix, it is calculated as ad - bc. If the determinant is zero, the system may have infinitely many solutions or no solution, meaning Cramer's Rule cannot be applied directly.

Alternative Methods for Solving Systems

When the determinant D equals zero, Cramer's Rule fails, so other methods like substitution, elimination, or matrix row reduction are used. These methods help determine if the system has no solution (inconsistent) or infinitely many solutions (dependent). Understanding these alternatives ensures a complete approach to solving linear systems.