Problem 75 - 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 two circles must have at least two real ordered-pair solutions

Step-by-Step Solution

Answer

The statement is false. The corrected statement is: A system of two equations in two variables whose graphs are two circles can have 0, 1, or 2 real ordered-pair solutions.

Step 1: Evaluate Statement

The initial statement to evaluate is that 'A system of two equations in two variables whose graphs are two circles must have at least two real ordered-pair solutions.'

Step 2: Understand Circle Equations

Firstly, it's important to understand that two circles in a plane can either intersect at two points, intersect at one point only, or have no intersection at all, depending on their relative positions and sizes.

Step 3: Decide True or False

According to the characteristics of circles, it's evident that the statement is false. For two circle equations, there could be 0, 1, or 2 real ordered-pair solutions, not 'at least two'.

Step 4: Correct The Statement

To correct the statement: 'A system of two equations in two variables whose graphs are two circles can have 0, 1, or 2 real ordered-pair solutions.'

Key Concepts

Circle Equations in Algebra

Algebra includes the study of circle equations, which are mathematical representations of circles on a coordinate plane. The general form of a circle equation is given by \( (x - h)^2 + (y - k)^2 = r^2 \), where \( h \) and \( k \) are the coordinates of the center of the circle, and \( r \) is the radius. It's essential to understand how these equations describe a circle's size and position.
By altering the values of \( h \) and \( k \) you change the location of the circle's center, and by changing \( r \) you adjust the circle's size. An important takeaway is that the variables \( x \) and \( y \) represent any point on the circle's circumference. This concept is foundational when exploring the interaction between two circles, such as determining their points of intersection.

Intersecting Circles Solutions

The intersection of circles is a fascinating topic in coordinate geometry. Two circles in a planar system may intersect in a maximum of two points. The points at which they intersect are known as the solutions to the system of circle equations.
These solutions can be found through algebraic methods, often involving setting the two circle equations equal to each other and solving for \( x \) and \( y \). Depending on the distance between the centers of the circles and the sum of their radii, the circles may:

• Not intersect at all (disjoint with no common points)
• Touch at exactly one point (tangential)
• Intersect at two points (secant)

The varying number of intersection points directly influences the number of possible solutions in the system of equations. Thus, when you're given a system of equations with circles, exploring these scenarios helps in understanding whether you'll have zero, one, or two real ordered-pair solutions.

Ordered Pair Solutions

Ordered pairs are pairs of numbers that represent the coordinates of points in a coordinate system, commonly denoted as \( (x, y) \). In the context of circle equations, ordered pair solutions are the intersection points' coordinates of two circles. These are the points that satisfy both circle equations simultaneously.
Finding these solutions is key in many geometry problems, and involves substituting one equation into another or using elimination methods. The number of solutions, as previously mentioned, can be zero, one, or two, depending on how the circles interact. If there are zero or one solutions, this reflects that the circles do not intersect or are tangent to each other. Two solutions indicate that the circles intersect at two distinct points, providing clear \( (x_1, y_1) \) and \( (x_2, y_2) \) pairs. In algebra, solving for these pairs requires careful calculation, cross-verification, and sometimes graphical analysis to ensure that the derived solutions actually reflect the geometric representation on the plane.