Lesson 1 Contents Example 1Number of Solutions Example 2Solve a System of Equations Example 3Write and Solve a System of Equations

Graphing Systems of Equations Two equations together are called a system of equations. A solution of a system of equations is an ordered pair of numbers that satisfies both equations. A system of two linear equations can have 0, 1, or an infinite number of solutions.

System of Equations InconsistentConsistent Parallel lines, no solutions Dependent (an infinite number of solutions, the equations graph the same line) Independent (exactly one solution, graphs intersect at a single point)

Example 1-1a 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 1-1a 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 1-1a 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.

Use the graph to determine whether each system has no solution, one solution, or infinitely many solutions. a. b. c. Example 1-1b Answer: one Answer: no solution Answer: infinitely many

Graphing Linear Equations Write the equation in Slope-Intercept Form: y = mx +b, where m is the slope and b is the y-intercept 1.Solve for y (get everything on the other side of the equal sign) 2.Begin with “b” – graph the y-intercept 3.From the y-intercept, use the slope (rise/run) to find your second point. 4.Draw your line and label with the equation.

Example 1-2a 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 1-2a 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 1-2b 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. Answer: one; (0, 3)

Example 1-2b Answer: no solution 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. b.

Example 1-3a 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 1-3a 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 1-3a 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.

Example 1-3a Check Answer: Tyler and Pearl walked for 3 hours.

Example 1-3b Alex and Amber are both saving money for a summer vacation. Alex has already saved $100 and plans to save $25 per week until the trip. Amber has $75 and plans to save $30 per week. In how many weeks will Alex and Amber have the same amount of money? Answer: 5 weeks number of weeks amount of money saved

Assignment Page 372: 15-22, odd