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Chapter 8: Problem 75

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.'

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Most popular questions from this chapter

An object moves in simple harmonic motion described by \(d=6 \cos \frac{3 \pi}{2} t,\) where \(t\) is measured in seconds and \(d\) in inches. Find: a. the maximum displacement b. the frequency c. the time required for one cycle. (Section \(5.8, \text { Example } 8)\)

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Solve: \(\sqrt{2 x-5}-\sqrt{x-3}=1 .\) (Section \(1.6,\) Example 4 )
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