1. Special Cases Involving Systems

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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)

12. 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)

13. 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.

14. 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

15. 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.