Question

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence.

If a consistent system as exactly two solution(s), it is said to be independent.

Short answer

The given statement is false and the corrected statement is “If a consistent system as exactly one solution, it is said to be independent.”.

Step-by-step solution

Step 1. State the concept of consistent and inconsistent solution.

A system of two linear equations can have one solution, an infinite number of solutions, or no solution.

1. If a system has at exactly one solution, more than one solution or infinitely many solutions, then it is said to be consistent.
2. If a system of equations has no solution, then it is said to be inconsistent.

Step 2. State the concept of independent and dependent solution.

If a consistent system has exactly one solution, it is independent and if a consistent system has an infinite number of solutions, it is dependent.

Step 3. State the conclusion.

Since, a system of two linear equations is independent if it has exactly one solution, therefore, the given statement “If a consistent system as exactly two solution(s), it is said to be independent” is false. The corrected statement will be:

If a consistent system as exactly one solution, it is said to be independent.