Problem 60 - Algebra and Trigonometry

Question

In applying Cramer's Rule, what should you do if \(D=0 ?\)

Step-by-Step Solution

Answer

In Cramer's Rule, if the determinant \(D=0\), it implies that the equations are either parallel or coincide. If the determinant of values is also 0, there are infinite solutions. However, if the determinant of the values is not 0, there are no solutions as the equations are parallel.

Step 1: Understanding Cramer's rule

Cramer's rule is a mathematical theorem used to solve a system of linear equations with the same number of equations as unknowns, and where the coefficient matrix (the determinant D) is non-zero.

Step 2: Identifying the special case

If the determinant D equals zero (i.e., D = 0), it means that the equations are either parallel or coincide. This is a special case in Cramer’s rule.

Step 3: Determining the solution in special case

In this case, if the determinant of coefficients equals zero (D = 0) and the determinants of values also equal zero, there are infinite solutions, as the equations coincide. If the determinant of coefficients equals zero (D = 0) but the determinants of values are not zero, then there are no solutions, as the equations are parallel.