A system of three linear equations in three variables is consistent and independent. How many solutions to the system exist?
A. none
B. one
C. three
D. infinitely many
Answer (1)
A consistent and independent system of three linear equations in three variables has exactly one solution. The terms 'consistent' implies at least one solution exists, and 'independent' implies there is a unique solution. Therefore, the system has o n e solution.
Explanation
Understanding the Problem A system of three linear equations in three variables is consistent and independent. This means the system has at least one solution (consistent) and the solution is unique (independent).
Reasoning A consistent system of equations has at least one solution. An independent system of equations has a unique solution. Therefore, a system that is both consistent and independent has exactly one solution.
Conclusion Therefore, the system has exactly one solution.
Examples
Consider a scenario where you're trying to determine the price of three different items (say, apples, bananas, and oranges) based on three different purchase combinations. If the equations representing these purchases are consistent and independent, it means there's exactly one set of prices that satisfies all three purchase scenarios. This is a practical application of solving a consistent and independent system of linear equations.
