Problem 69 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When using Cramer's Rule to solve a linear system, the number of determinants that I set up and evaluate is the same as the number of variables in the system.

Step-by-Step Solution

Answer

The statement does not entirely make sense. In using Cramer's Rule, the number of determinants evaluated is always one more than the number of variables because an additional determinant for the system itself is also calculated.

Step 1: Understanding Cramer's Rule

Cramer's Rule is a mathematical theorem used for solving systems of linear equations with as many equations as variables. It uses the determinant of the matrix of the system and replaced columns corresponding to each variable.

Step 2: Relevance with the statement

According to Cramer's Rule, for a system of 'n' variables, 'n' determinants should be calculated: One for the system itself (the determinant of the coefficient matrix), and 'n-1' for the determinants obtained by replacing each column with the constants from the equations. So in a way, the number of evaluated determinants is always one more than the number of variables.

Step 3: Conclusion

So, the statement doesn't entirely make sense because while it's true that a determinant is calculated for each variable, an additional determinant for the system itself is also calculated. So, the total number of determinants calculated is always one more than the number of variables in the system.