lesson 7-1 graphing systems of equations. transparency 1 click the mouse button or press the space...
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Lesson 7-1
Graphing Systems of Equations
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Objectives
• Determine whether a system of linear equations has 0, 1, or infinitely many solutions
• Solve a system of equations by graphing
Vocabulary
• System of equations – two or more equations
• Consistent – a system of equations that has at least one ordered pair that satisfies both equations
• Inconsistent – a system of equations with no ordered pair that satisfies both equations
• Independent – a system of equations with exactly one solution
• Dependent – a system of equations that has an infinite number of solutions
System of Equalities
• Solutions of two linear equations result in:
y
x
y
x
y
x
No Solutions One Solution Infinite Solutions
Because (graphically): Lines are parallel Lines Intersect Same Line
Example 1a
Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions.
Answer: Since the graphs of andare parallel, there are no solutions.
Example 1bUse the graph to determine whether the system has no solution, one solution, or infinitely many solutions.
Answer: Since the graphs of andare intersecting lines, there is one solution.
Example 1cUse the graph to determine whether the system has no solution, one solution, or infinitely many solutions.
Answer: Since the graphs of andcoincide, there are infinitely many solutions.
Example 2a
The graphs of the equations coincide. There are infinitely many solutions of this system of equations.
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.
Answer:
Example 2b
The graphs of the equations are parallel lines. Since they do not intersect, there are no solutions of this system of equations.
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.
Answer:
Example 3
Bicycling Tyler and Pearl went on a 20-kilometer bike ride that lasted 3 hours. Because there were many steep hills on the bike 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 andthe number of hours they walked. Write a
system of equations to represent the situation.
Example 3 contEquations
The number ofhours riding plus
the number ofhours walking equals
the total number of hours of the trip.
The distancetraveled riding plus
the distancetraveled walking equals
the total distance of the trip.
r + w = 3
12r + 4w = 20
Example 3 contGraph the equations and .
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.
Answer: Tyler and Pearl walked for 3 hours.
Summary & Homework
• Summary:
• Homework: – Pg 372 16-36 even
Graph Reveals
Intersecting Lines
Same Line
Parallel Lines
Solutions One Infinitely many none
Terminology Consistent and independent
Consistent and dependent
inconsistent