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.
4x + 3y = -7
2x + 3y = -11

검증된 단계별 안내

Write the system of equations in standard form: \\ \(4x + 3y = -7\) and \(2x + 3y = -11\).

Identify the coefficient matrix and calculate the determinant \(D\(: \\ \)D = \begin{vmatrix} 4 & 3 \\ 2 & 3 \end{vmatrix} = (4)(3) - (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} -7 & 3 \\ -11 & 3 \end{vmatrix} = (-7)(3) - (-11)(3)\).

Calculate the determinant \(D_y\( by replacing the second column of the coefficient matrix with the constants: \\ \)D_y = \begin{vmatrix} 4 & -7 \\ 2 & -11 \end{vmatrix} = (4)(-11) - (2)(-7)\).

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 the system does not have a unique solution and you should use another method such as substitution or elimination to find the solution set.

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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 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 matrix row reduction (Gaussian elimination). These methods help determine whether the system has no solution, infinitely many solutions, or a unique solution by analyzing the equations' consistency.

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. 4x + 3y = -7 2x + 3y = -11