A system of three linear equations in three variables is consistent and dependent. How many solutions to the system exist?
A. none
B. one
C. three
D. infinitely many
Answer (2)
A consistent system has at least one solution.
A dependent system has infinitely many solutions.
A consistent and dependent system has infinitely many solutions.
Therefore, the system has infinitely many solutions.
Explanation
Understanding the Problem The problem states that we have a system of three linear equations in three variables that is consistent and dependent. We need to determine how many solutions exist for such a system.
Consistent and Dependent Systems A consistent system of equations is one that has at least one solution. A dependent system of equations is one where the equations are not independent, meaning that at least one equation can be written as a linear combination of the others. In the case of a linear system, dependency implies that there are infinitely many solutions.
Combining the Information Since the system is consistent, it has at least one solution. Since the system is dependent, the equations are not independent, which means there are infinitely many solutions.
Final Answer Therefore, a system of three linear equations in three variables that is consistent and dependent has infinitely many solutions.
Examples
Consider a scenario where you're mixing three ingredients to create a product, and you have three equations representing the constraints on the amounts of each ingredient. If the system is consistent and dependent, it means there are infinitely many combinations of the ingredients that satisfy the constraints. This allows for flexibility in production while still meeting the required conditions. Understanding consistent and dependent systems helps in optimizing resource allocation and finding multiple solutions to real-world problems.
A system of three linear equations in three variables that is consistent and dependent has infinitely many solutions. This occurs because consistent systems have at least one solution, while dependent systems provide an infinite number of solutions. Thus, the chosen option is D: infinitely many.
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