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 SolutionsOne SolutionInfinite 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 ofand are parallel, there are no solutions.

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

Example 1c Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. Answer: Since the graphs ofand coincide, 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 and the number of hours they walked. Write a system of equations to represent the situation.

Example 3 cont Equations The number of hours riding plus the number of hours walking equals the total number of hours of the trip. The distance traveled riding plus the distance traveled walking equals the total distance of the trip. r+w= 3 12r+4w4w= 20

Example 3 cont Graph the equationsand. 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 even Graph Reveals Intersecting Lines Same Line Parallel Lines SolutionsOneInfinitely manynone TerminologyConsistent and independent Consistent and dependent inconsistent