Problem 73 - College Algebra

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 a circle and a line can have four real ordered-pair solutions

Step-by-Step Solution

Answer

The given statement is false. The correct statement would be 'A system of two equations in two variables whose graphs are a circle and a line can have at most two real ordered-pair solutions.'

Step 1: Visualizing the Intersections

Visualizing a line and a circle, it should be clear that a line can at most intersect a circle at two points. To imagine this, draw a circle and a line that goes through it, and observe that there are only two points where the line crosses the boundary of the circle.

Step 2: Analyzing the statement

The statement says that there can be four real ordered-pair solutions. This refers to the intersections between the line and the circle. By our first step analysis, it can now be concluded that the statement is false, as there can be at most two real ordered-pair solutions, not four.

Step 3: Correcting the statement

The correct version of the statement would be 'A system of two equations in two variables whose graphs are a circle and a line can have at most two real ordered-pair solutions.'

Key Concepts

systems of equations

At the heart of many mathematical problems lies the concept of systems of equations. When we have two or more equations working together, depending on each other, it forms a system. For example, a system of equations might involve a circle and a line. These equations work together to describe a particular scenario or model. For instance, when a line intersects a circle, their system of equations will display certain solutions.

• Each equation represents a different shape or line on a graph.
• The solution to the system is the point or points where these shapes or lines intersect.
• It's all about finding common solutions that satisfy both (or all) equations involved.

Understanding systems of equations involves recognizing the relationships they depict. Whether they show shapes, numbers, or another real-world scenario, mastering them helps solve complex puzzles clearly and logically.

circle and line intersection

Understanding the interaction between a circle and a line can open up a world of geometric fun! When we talk about a circle, we picture a round, perfectly symmetrical shape. A line, in contrast, stretches infinitely in both directions.

• The line can meet the circle at zero, one, or two points.
• If it meets the circle at one point, we call it a tangent.
• If it doesn’t meet the circle at all, it's because the line doesn’t reach the circle.
• At most, a line can intersect a circle at two distinct points.

In mathematical terms, an intersection is simply where two shapes share a common point. When we deal with circles and lines, intersections tell us where exactly the line "cuts through" the circle. This principle is crucial for solving many geometry and algebra problems.

real ordered-pair solutions

When discussing the intersection of shapes such as lines and circles, the concept of real ordered-pair solutions becomes vital. But what exactly does this mean? Real ordered-pair solutions are the coordinates of the points where the geometrical figures intersect.

• An ordered pair is a set of two numbers written in a specific sequence, usually as (x, y).
• In geometric terms, these numbers represent a point on a graph.

Think of these pairs as precise "addresses" that tell you where the two shapes meet on the grid. In our example with the circle and line, each intersection gives us one ordered-pair solution. That's why, when we say the maximum number of real ordered-pair solutions a circle and line can have is two, we mean there can be up to two distinct points (x, y) where the line meets the circle.