Lesson 7 1 Graphing Systems of Equations Click

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: No Solutions One Solution y Infinite Solutions y x Lines are parallel y x Because (graphically): Lines Intersect x Same Line

Example 1 a Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. Answer: Since the graphs of and are parallel, there are no solutions.

Example 1 b Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. Answer: Since the graphs of and are intersecting lines, there is one solution.

Example 1 c Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. Answer: Since the graphs of and coincide, there are infinitely many solutions.

Example 2 a 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: The graphs of the equations coincide. There are infinitely many solutions of this system of equations.

Example 2 b 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: The graphs of the equations are parallel lines. Since they do not intersect, there are no solutions of this system of equations.

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 and the number of hours they walked. Write a system of equations to represent the situation.

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

Example 3 cont Graph 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: Graph Reveals Solutions Intersecting Lines Same Line Parallel Lines One Infinitely many none Consistent and dependent inconsistent Terminology Consistent and independent • Homework: – Pg 372 16 -36 even