Special Cases Involving Systems

Types of Systems Consistent Inconsistent Two lines intersect in at least one point Have at least one solution Inconsistent Parallel lines—never intersect No solution Slopes are the same, y-intercepts are different

Types of Consistent Systems Independent Intersect in one point Has exactly one solution The graph has two lines that intersect at one point Dependent Intersect in infinitely many points Has infinitely many solutions The graph has two lines that are graphed on top of each other. They coincide

Consistent and Independent Consistent and Dependent Inconsistent Classification Consistent and Independent Consistent and Dependent Inconsistent Number of Solutions Exactly one Infinitely many None Description Different slopes Same slope, same y-intercept Same slope, different y-intercept Graph

Example 1. What is the best way to determine whether the system has no solution, one solution, or infinitely many solutions? y = –x + 1 y = –x + 4 Answer: Graphing since both equations are in slope intercept form. Since the graphs are parallel, there are no solutions. Inconsistent

Example 2 What is the best way to determine whether the system has no solution, one solution, or infinitely many solutions. 3x – 3y = 9 y = –x + 1 Answer: Substitution since one equation is in slope intercept form. Since the graphs are intersecting lines, there is one solution. Consistent and independent

Example 3 Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. 2y + 3x = 6 y = x – 1 A. one B. no solution C. infinitely many D. cannot be determined

Example 4. Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. y = x + 4 y = x – 1 A. one B. no solution C. infinitely many D. cannot be determined

Example 5. Determine the best way to determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 2x – y = –3 8x – 4y = –12 Answer: Elimination since both equations are in standard form. The graphs coincide. There are infinitely many solutions of this system of equations. Consistent and dependent

Example 6. What the best method to determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. x – 2y = 4 x – 2y = –2 Answer: Elimination since both equations are in standard form. The graphs are parallel lines. Since they do not intersect, there are no solutions of this system of equations. Inconsistent

Example 7. Graph the system of equations Example 7. Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. A. one; (0, 3) B. no solution C. infinitely many D. one; (3, 3)

Example 8. Determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. A. one; (0, 0) B. no solution C. infinitely many D. one; (1, 3)

BICYCLING Tyler and Pearl went on a 20-kilometer bike ride that lasted 3 hours. Because of the steep hills on the ride, they had to walk for most of the trip. Their walking speed was 4 kilometers per hour. Their riding speed was 12 kilometers per hour. How much time did they spend walking? Words You have information about the amount of time spent riding and walking. You also know the rates and the total distance traveled. Variables Let the number of hours they rode and the number of hours they walked. Write a system of equations to represent the situation.

Equations r + w = 3 12r + 4w = 20 The number of hours riding plus the number of hours walking equals the total number of hours of the trip. r + w = 3 The distance traveled riding plus the distance traveled walking equals the total distance of the trip. 12r + 4w = 20

Graph the equations. r + w = 3 and 12r + 4w = 20. The graphs appear to intersect at the point with the coordinates (1, 2). Check this estimate by replacing r with 1 and w with 2 in each equation.