Question

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

If a consistent system has an infinite number of solutions, it is said to be inconsistent.

Short answer

The given statement is false. The corrected statement is:

If a consistent system has an infinite number of solutions, it is said to be dependent system.

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. Explanation of the statement.

Since, a system of two linear equations can have the following three solution conditions:

• One solution
• An infinite number of solutions
• No solution.

That’s why the systems of equations are classified by the set of solutions to satisfy the equations.

The system with the infinite number of solutions is called as dependent system. When there is no solution for the given consistent system then it is said to be inconsistent system.

Step 3. Write the correct statement.

The given statement is wrong. The corrected statement is:

If a consistent system has an infinite number of solutions, it is said to be dependent system.