Problem 75 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A system of two equations in two variables whose graphs are two circles must have at least two real ordered-pair solutions.

Step-by-Step Solution

Answer

The statement is false. A system of two equations in two variables whose graphs are two circles can have zero, one, or two real ordered-pair solutions.

Step 1: Understand the statement

The statement given is: 'A system of two equations in two variables whose graphs are two circles must have at least two real ordered-pair solutions.' It implies that whenever two circles intersect or coincide, there should be at least two points of intersection which are real ordered pairs.

Step 2: Analyze the situation of the circles

Two circles can intersect at two points, one point, or no point. The situation of no intersection translates to no real ordered-pair solution. The intersection at a single point gives one real ordered-pair solution. Therefore, it's not always true that two circles must intersect at least twice.

Step 3: Correct the statement

The current statement is false. The correct statement would be: 'A system of two equations in two variables whose graphs are two circles can have zero, one, or two real ordered-pair solutions.'