# Name Date PD. Systems of Equations and Inequalities

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1 Name Date PD Sstems of Equations and Inequalities

2 Sstems of Equations Vocabular: A sstem of linear equations is A solution of a sstem of linear equations is Points of Intersection (POI) are the same thing as the solutions of a sstem. No solution means A sstem of equations has infinitel man solutions when

3 Solve b graphing: Sstems with No solutions.): Sstems with Infinitel Man solutions.) E. ) Tr-It)

4 E. ) Tr-It) E. ) Tr-It)

5 Graphing Standard Form Sstems: E. ) 0 Tr-It) Tr-It) Tr-It)

6 Practice Graph the two linear equations and find the solution.....

7 .. Find the solution to the given sstems...

8 Linear Inequalities/ Sstems of Inequalities Steps:. Write the equation in = m + b form (if necessar).. Graph the equation.. Draw a solid line if the inequalit is or.. Draw a dashed line if the inequalit is < or >.. Pick a point that is not on the line to use as a test point. The origin (0, 0) is usuall good to use if it is not on the line.. If the point makes the inequalit true, shade that side of the line. If the point does not make the inequalit true, shade on the opposite side of the line. Practice: #. Graph. #. Graph.

9 #. Graph. #. Graph. #. Graph #. Graph.

10 #. Graph. #. Graph #9. Graph #0. Graph 9 0

11 Practice #. Graph #. Graph

12 #. Graph #. Graph #. Graph #. Graph

13 Write a sstem of inequalities for the dark region of each graph. #. # Word Problems. Suppose our class is raising mone for the Red Cross. You make \$ on each basket of fruit and \$ on each bo of cheese that ou sell. How man items of each tpe must ou sell to raise more than \$00?

14 . JoAnn likes her job as a bab-sitter, but it pas onl \$ per hour. She has been offered a job as a tutor that pas \$ per hour. Because of school, her parents onl allow her to work a maimum of hours per week. How man hours can JoAnn tutor and bab-sit and still make at least \$ per week? How much does she make? Math Club members want to advertise their fundraiser each week in the school paper. The know that a frontpage ad is more effective than an ad inside the paper. The have a \$0 advertising budget. It cost \$ for each front-page ad and \$ for each inside-page ad. The club wants to advertise at least 0 times. a) Write and graph a sstem of inequalities to model the number of advertisements the club can purchase to sta under budget. Be sure to label all parts of our graph. b) State one solution that would work. How much mone will remain in the club s budget?

15 . On a canoe trip, Rita paddled upstream (against the current) at an average speed of mi/h relative to the riverbank. On the return trip downstream (with the current), her average speed was mi/h. Find Rita s paddling speed in still water and the speed of the river s current.. A light plane flew from its home base to an airport miles awa. With a head wind, the trip took. hours. The return trip with a tail wind took. hours. Find the average airspeed of the plane and the average wind speed.

16 Practice. Suppose ou intend to spend no more than \$0 buing books. Hardcover books cost \$ and paperback cost \$. How man books of each tpe can ou bu? Miss Jones wants to bu at least 0 books. Each paperback book costs an average of \$0 and each hardcover book costs an average of \$0. Miss Jones is planning to spend less than \$0 on books. c) Write and graph a sstem of inequalities to model the number of books she can purchase to sta under budget. Be sure to label all parts of our graph. d) State one solution that would work. How much did she spend on hardcover books?

17 . The science club wants to bu t-shirts for the organization. The have a budget of \$0. Plain white t-shirts cost \$ and red t-shirts with writing costs \$. There are 0 students in the club but all are not sure if the are going to purchase a t-shirt. a) Write and graph a sstem of inequalities to model the number of t-shirts the club can purchase to sta under budget. b) State one solution that would work. How man more white t-shirts could the bu? Eplain It took hours for a plane, fling against the wind, to travel 900 miles from Alabama to Minnesota. The ground speed of the plane is 00 miles per hour. On the return trip, the flight took onl hours with a ground speed of 0 miles per hour. During both flights the speed and the direction of the wind were the same. Find the speed of the plane in still air and the speed of the wind.. You are traveling to a concert. To arrive on time, ou plan to travel at an average speed of miles per hour. For the first two hours of the trip ou travel at 0 mph, then ou realize ou need to speed up to get there on time so ou finish the trip traveling at mph. Use the following model to write the sstem to determine how long ou traveled at mph and how long the total distance was to get there. Distance at 0 mph Distance at mph Total Distance Average speed Total Time Total Distance

18 Substitution Graph the following sstem of equations, find the solution.. Tr It! Solve Algebraicall a. 0 b. c.

19 Eample - Word Problems Tre s online music club charges a monthl rate of \$0 plus \$0.0 per song download. Deb s online music club charges a monthl rate of \$ plus \$0.0 per song download. For what number of songs will the monthl charge be the same for both clubs? How much will it cost? Tr It! One car model costs \$,000 and costs an average of \$0.0 per mile to maintain. Another car model costs \$,000 and costs an average of \$0.0 per mile to maintain. If one of each model is driven the same number of miles, after how man miles would the total cost be the same?

20 Eample Substitution. Tr It! Tr It! Solve Algebraicall..

21 Etra Practice - Solve the following sstems

22 .. Confirm the answer algebraicall: Solve the sstem of equations algebraicall, find the solution.... Suppose a video store charges nonmembers \$ to rent each video. A store membership costs \$ and members pa onl \$.0 to rent each video. For what number of videos is the cost the same?. Suppose our club is selling candles to raise mone. It costs \$00 to rent a booth from which to sell the candles. If the candles cost our club \$ each and are sold for \$ each, how man candles must be sold to equal our epenses?

23 Elimination You have alread developed some useful strategies for solving a simple linear equation like + = 0. You know that ou can add or subtract the same quantit on both sides and preserve equalit. The same is true for multiplication or division. These ideas, called the Properties of Equalit, can help ou develop another method for solving linear equations. This method involves combining separate linear equations (through the four basic operations) into one equation with onl one variable. Other names for this method are Combination, Addition, or Elimination.

24 Eamples: Tr It! a. b. 0 Eamples:.. 9 Tr It! a. b. 0 9

25 Now let s investigate some other sstems that involve other uses of the elimination method Tr It! a. 9 9 b. Practice Solve b elimination. Show our work... 0

26 Word Problems Tr It! a. 9 b.

27 Eamples:. The sum of two numbers is. The sum of the greater number and twice the smaller number is. Find the numbers.. Suppose a band at another school sells erasers for \$ per package and pencils for \$ per package. The band sells 0 packages in all and earns a total of \$9. Write a sstem of equations to find the number of each tpe of package sold. Tr It! a. Grandma s Baker sells single crust apple pies for \$.99 and double crust cherr pies for \$0.99. The total number of pies sold on a bus Frida was thirt-si. If the amount collected for all the pies that da was \$., how man of each tpe of pies was sold?

28 b. A promoter priced tickets to a concert as follows: \$ when purchased in advance and \$0 when purchased at the door. The total number of tickets purchased was, and ticket sales totaled \$9,. How man tickets were sold at the door? Practice Solve b elimination. Show our work

29 Use a sstem of equations to model each situation. Solve b an method.. The sum of two numbers is 0. The sum of the greater number and three times the lesser number is. Find the numbers.. Shopping at Savers Mart, Lisa bus her children four shirts and three pairs of pants for \$.0. She returns the net da and bus three shirts and five pairs of pants for \$.00. What is the price of each shirt and each pair of pants?. An amusement park charges admission plus a fee for each ride. Admission plus two rides costs \$0. Admission plus five rides cost \$. What is the charge for admission and the cost of a ride?

30

### Solve each system by graphing. Check your solution. y =-3x x + y = 5 y =-7

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