Problem 1 - Beginning and Intermediate Algebra

Question

If a nonlinear system consists of equations with the following graphs, a) sketch the different ways in which the graphs can intersect. b) make a sketch in which the graphs do not intersect. c) how many possible solutions can each system have? circle and line

Step-by-Step Solution

Answer

A circle and a line can intersect in three different ways: (1) no points in common (don't intersect), (2) one point in common (tangent to the circle), and (3) two points in common (secant to the circle). To create a sketch where the graphs do not intersect, draw a circle and a line that is far enough from the circle that they do not touch or intersect at any point. The maximum number of possible solutions that such a system can have is 2.

Step 1: Understand the intersection of a circle and a line

To analyze possible intersections between a circle and a line, recall that a circle is a set of points that are equidistant from the center, and a line is a straight path that extends indefinitely in both directions. The distance between the center of the circle and the line determines the type of intersection.

Step 2: Sketch the different ways in which the graphs can intersect

A circle and a line can intersect in three different ways: (1) no points in common (don't intersect), (2) one point in common (tangent to the circle), and (3) two points in common (secant to the circle). Sketch these possibilities: 1. A single point of intersection (tangent scenario): Draw a circle and a line that just touches the circle at one point. 2. Two points of intersection (secant scenario): Draw a circle and a line that passes through the circle, intersecting it at two points.

Step 3: Sketch a scenario where the graphs do not intersect

To create a sketch where the graphs do not intersect, draw a circle and a line that is far enough from the circle that they do not touch or intersect at any point.

Step 4: Determine the maximum number of possible solutions

As mentioned in Step 2, the circle and line can intersect in three scenarios: no intersection, single-point intersection, and two-point intersection. As such, the maximum number of possible solutions that such a system can have is 2. Thus, the nonlinear system consisting of a circle and a line can have at most 2 possible solutions.