Algebra 1 Unit 3 Systems of Linear Equations and ... · Unit 3: Systems of Equations and Inequalities 16 Lesson 3-3: Solving Systems by Substitution Learning Target: I can solve systems

Unit3:SystemsofEquationsandInequalities

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Algebra 1 Unit 3 Systems of Linear Equations and Inequalities

Table of Contents

Title Page # Formula Sheet…………………………………………………………………….2 Lesson 3-1 Graphing Systems of Equations.……………….…..……….....3 Lesson 3-2 Graphing Systems of Inequalities ……………….…………… 10 Lesson 3-3 Solving Systems by Substitution..……………………..….… 16 Lesson 3-4 Solving Systems by Elimination.………………………….…. 23 Lesson 3-5 Arithmetic Sequences.………………………………………… 32 Unit 3 Study Guide…………………………………………………………….42 Glossary…………………………………………………………………………48

This packet belongs to:

__________________________________________________________


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Lesson 3 – 1: Graphing Systems of Equations

Learning Target: I can solve systems of equations by graphing. REI.6 & REI.11

• Asystemoflinearequationsis

______________________________________________

______________________________________________________________________________

• Asolutionofasystemoflinearequationsis____________________________________

______________________________________________________________________________Thisissometimesreferred

toasthe_________________________________.

• Nosolutionmeans___________________________________________________________________

_________________________________________________________________________________________________

• Asystemofequationshasinfinitelymanysolutionswhen:________________________

__________________________________________________________________________________________________

Write your Questions here!

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Examples:

1.)

2.)

Solution(s):____________________ Solution(s):____________________

3.)

4.)


îíì

-=+=2323

xyxy

ïïî

ïïí

ì

+-=

+-=

343

343

xy

xy

îíì

=-=-yxyx

362

îíì

-==62

yx


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1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y


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Sometimes,youmayseesystemsofequationswritteninaslightlydifferentway.Insteadofseeingthesystemwrittenastwoseparateequations,youmayoccasionallyseethemcombinedintooneequation.Wecanstillsolvethesebygraphing,wejusthave_______________________________________________________________________________.Example5Solvetheequationbygraphing

−2𝑥 + 3 = 5 − 3𝑥f(x)=________________________g(x)=_______________________Solution(s):________________


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Practice 3-1: Graphing Systems of Equations Solve each system by graphing. Give the solution(s) on the line provided.

1) 2)


3) 4))3𝑥 − 3𝑦 = −62𝑥 − 𝑦 = −1


ïî

ïíì

-=

--=

4

121

xy

xy

îíì

-==43

xy

ïïî

ïïí

ì

=+-

=+-

yx

yx

212

223

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y



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5) )4𝑥 + 2𝑦 = 4−3𝑥 + 𝑦 = 2 6)


7)

8)


ïî

ïíì

=

+=

yx

xy

21

32

ïî

ïíì

+-=

-=

23152

xy

xy y = 4xx = −2"#$

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y


1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

–1


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Graphandsolvethefollowingsystemsofequations.9)𝟏

𝟐𝒙 + 𝟑 = 𝒙 + 𝟒 10)𝟑𝒙 + 𝟔 = 𝒙 − 𝟐

f(x)=_______________________f(x)=_______________________g(x)=_______________________ g(x)=_______________________solutions=__________________ solutions=___________________

11)Whichisabetterdealona$50item:

o 45%offtheoriginalpriceo 30%offwithanadditional20%offattheregister.

12)Usethegraphstodeterminethey-interceptandslope,thenwritetheequation.Towriteequationsoflinesweoftenuseslope-intercept.

Slope-interceptformis____________________,wheremisthe_____________andbisthe____________.


This WILL be

on your mastery check!

Spiral Practice


9

13.GraphtheInequality

f(x)>–½x–5 y≤–5

ReviewyourpracticeandnotestopreparefortheMasteryCheck.


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Lesson 3-2: Graphing Systems of Inequalities Learning Target: I can show the solution set of a system of inequalities by graphing on the coordinate plane. A.CED.3 Recall that when graphing an inequality, the solution set consists of an entire shaded region rather than just the line. The solution set to a system of inequalities requires you to graph both inequalities and look for the ____________ of the shaded regions.

Identify2orderedpairsthatareinthesolutionset:

Identify2orderedpairsthatarenotinthesolutionset:

A system of equations with parallel lines _______________________. Parallel lines in a system of inequalities are different. In order to determine the solution set, you have to graph and inspect the shaded regions.

y < x + 4

y > x − 2"#$

y ≥ 2x + 4y > 2x − 1#$%

y > −3x + 5y < −3x − 3"#$


y xy x≤ +> − −

#$%

2 36

All solutions are in this double shaded area.

𝒚 ≤ 𝟐𝒙 + 𝟑 𝒚 > −𝒙− 𝟔

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YouTry: For each system below, give two ordered pairs that are solutions and two that are not solutions.

Systems of inequalities can be used to help us see certain restrictions in various scenarios. Example 1: Write and graph a system of inequalities to model the situation. Ted wants to throw a birthday party at Yonah Bowl. Yonah Bowl charges $10 a person to bowl and $15 a person to skate. Ted has no more than $180 to budget for his party. His guests can choose to either bowl or skate. Ted wants to invite at least ten friends. Write an inequality that models his situation and display it on the graph below. What are two possible combinations of guests that Ted can have?


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Youtry:Marshall can work no more than twenty hours a week at his two jobs. He makes $5 per hour babysitting and $6 per hour tutoring. He needs to earn at least $90 per week to cover his expenses. Write a system of inequalities that models the situation and graph the system on the provided graph. What are two possible combinations that Marshall could work?



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Practice3-2SolvingSystemsByGraphing Graph each system of linear inequalities.

1. 2. 7𝑦 < 𝑥

2𝑥 + 𝑦 > 1

3. 4.

7.

36

y xy x> -ì

í ³ - +î

33

y xy x£ -ì

í > +î2 22 3

y xy x> -ì

í £ +î

15

y xy x> - -ì

í > - -î



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7. )𝑥 − 𝑦 > −1−𝑥 − 𝑦 < 5 8.9

𝑦 ≤ :;𝑥 + 5

𝑦 ≤ −;<𝑥 + 3

Write and graph a system of inequalities to model the situation. 9. Barney is selling candy bars and sodas to make money for summer vacation. The candy bars cost $2 and the sodas cost $3. He needs to make at least $60. Barney knows he will sell more than 10 candy bars. Write inequalities to represent the income from the snacks sold and the number of candy bars sold. Give two combinations that would Barney could buy.



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10.Lily is buying wings and hot dogs for a party. One package of wings costs $8. Hot dogs cost $5 per package. She must spend less than $40. Lily knows she will be buying at least 4 pounds of hot dogs. Write a system of inequalities and graph it on the provided graph. Give two possible combinations that Lily can purchase.

Solve the following equations 11) 12)=>?@:>

=>= −50

13)Whatisthepriceperitemifyoupurchasean$8.99productatbuy2,getonefree?

14)Thesquarewithsidelength andanequilateraltrianglewithsidelength havethesameperimeter.Whatisthevalueofx?


( )3+x( )13 -x

This WILL be on your mastery check!

Spiral Practice


𝟐(𝟒 − 𝟐𝒓) − 𝟓 = −𝟐(𝒓 + 𝟓) + 𝟖𝒓


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Lesson 3-3: Solving Systems by Substitution

Learning Target: I can solve systems of equations using the substitution method. A.REI.5

There are times when graphing to solve a system of equations isn’t an option. Luckily, we have algebraic methods to save the day! Remember, the Substitution Property of Equality says that if we know two values are equal, we can replace them with each other in any equation or expression. This property can be extremely helpful when solving systems of equations. Sometimes, we may be given two equations that both have the same variable isolated.

Example 1:

You try:

1. )𝑦 = 4𝑥 − 10𝑦 = 6𝑥 − 14 2. ) 𝑥 = 5𝑦 + 21

𝑥 = −3𝑦 − 19

îíì

+=+-=53

xyxy


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Vocabulary: - Substitution


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Sometimes, we may be given a system of equations where just one equation has an isolated variable.

Example 2: Example 3:

You try:

1. )−3𝑥 − 8𝑦 = 9𝑦 = −7𝑥 − 21 2. ) 𝑥 = 𝑦 − 1

3𝑦 − 6𝑥 = −18

Finally, we may be given a system of equations where neither equation has an isolated variable. Don’t worry though, we can still use substitution!

Example 4:

y = −2x −14x + 2y =12"#$

x = −2y+ 52x + 4y =10"#$

4y+ x =− 43x + 5y = 2"#$


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You try:

1. )𝑥 − 3𝑦 = −36𝑥 − 𝑦 = 16 2. )2𝑥 + 7𝑦 = 11

3𝑥 + 𝑦 = 7

Hearken back to Lesson 3-1. We learned that there were three possible scenarios for a system of equations: one solution, no solutions, and infinitely many solutions. We discussed how to tell from a graph which scenario applied. Now we will look at how to tell algebraically.

Example 5:) 𝑦 = −7𝑥 + 2

14𝑥 + 2𝑦 = 7 Example 6: ) 𝑦 = 5𝑥 + 13−10𝑥 + 2𝑦 = 26

Example 7: )2𝑥 + 3𝑦 = −9𝑦 = −3𝑥 − 3

I


Begin3-3Video4

Whatyouthink. Whatyousee. Whatitmeans.TypicalResult x=3 OnesolutionWell,duh… x=xor2=2 InfinitelymanysolutionsNoitdoesn’t… 2=3 Nosolution


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Example 8: SupposeFCAissellingcandlestoraisemoney.Itcosts$100torentaboothfromwhichtosellthecandles.IfthecandlescostsFCA$1eachandaresoldfor$5each,howmanycandlesmustbesoldtoequalyourexpenses.

You Try: 1.SupposeAmazoniacharges$4torentadigitalvideo.AnAmazoniaCompositemembershipcosts$21andmemberspayonly$2.50torenteachvideo.Forwhatnumberofvideosisthecostthesame?


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Practice3-3SolvingSystemsBySubstitutionSolve by substitution. Tell whether the system has no solution, one solution or infinitely many solutions.

1. 2.

3. 4.

5. 6.

7. 8.

9.

10.

îíì

=+=xyxy34 y = −2x −1

4x + 2y =12"#$

îíì

-=+=11555

xyxy −2x + y = −7

2x − y =14"#$

îíì

-=+--=

6363yxxy x + 2y = 200

x − y = 50"#$

îíì

+==+1232

xyyx

ïî

ïíì

=-

=

14623

yx

xy

y = 13x − 4

x = −6

"

#$

%$ îíì

+=+=2264

xyxy



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Write a system of equations to model each scenario. Then solve the system. Make sure to give your answer in the context of the problem. 11.PatrickandPhiliphavedecidedtowatchalltheepisodesofWalkingDeadonNetflix.Patrickhasalreadywatched12episodesandwillcontinuetowatchtheshowatarateof1episodeperday.Phillip,whohasn'tyetseentheshow,willstartwatching4episodesperday.OncePatrickandPhillipgettothepointwheretheyhavewatchedthesamenumberofepisodes,theyplantofinishtheseriestogether.Howlongwillthattake?Howmanyepisodeswilleachhavewatchedatthatpoint?12. Car A costs $10,000 to buy and costs an average of $0.25 per mile to maintain. Car B costs $15,000 to buy and costs an average of $0.15 per mile to maintain. After how many miles would the total cost for each car be the same?

13. One car model costs $12,000 and costs an average of $0.10 per mile to maintain. Another car model costs $14,000 and costs an average of $0.08 per mile to maintain. If one of each model is driven the same number of miles, after how many miles would the total cost be the same?



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Solve each system below using either substitution or graphing. Graphs are provided at the bottom of the page, should you elect to solve by graphing.

14)J𝑦 = −𝑥 − 3𝑦 = =

:𝑥 + 3 15) )𝑦 = 2𝑥 − 10

𝑦 = 5𝑥 − 16

16) )5𝑥 + 6𝑦 = 177𝑥 + 𝑦 = 9 17) 9

𝑦 = <;𝑥 − 2

𝑦 = =;𝑥 + 3



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Lesson 3 – 4: Solving Systems by Elimination Learning Target: I can solve systems using the elimination method. A.REI.5 & A.REI.6

You have already developed some useful strategies for solving a simple linear equation like 3x + 5 = 10. You know that you can add or subtract the same quantity on both sides and preserve equality. The same is true for multiplication or division. These ideas, called the__________________, can help you develop another method for solving systems of equations. This method involves combining separate linear equations (through the four basic operations) into one equation with only one variable. You may hear this referred to as ____________________________.

Example 1: ) 𝑥 + 𝑦 = 3𝑥 − 𝑦 = −9 Example 2: ) 𝟑𝒙 + 𝟓𝒚 = 𝟖

−𝟑𝒙− 𝒚 = −𝟒

You Try:

1) )2𝑥 + 2𝑦 = −20−𝑥 − 2𝑦 = 19 2. )−5𝑥 + 7𝑦 = −17

4𝑥 − 7𝑦 = 15

Example 3: ) 6𝑥 + 3𝑦 = 0

−3𝑥 + 3𝑦 = 9


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Yourpropertiesofequalitiescanbe

foundinAppendixAinyourUnit1packet!


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You Try:

1) )– 𝑥 + 𝑦 = −92𝑥 + 𝑦 = 3 2) )5𝑥 + 3𝑦 = −30

5𝑥 + 9𝑦 = −30

Example 4:)2𝑥 + 𝑦 = 60

𝑥 + 2𝑦 = 75 Example 5: ) 4𝑥 + 3𝑦 = 25𝑥 + 2𝑦 = 13

You Try: 1) )5𝑥 + 4𝑦 = −4

4𝑥 + 6𝑦 = 8 2) )−2𝑥 − 2𝑦 = 24𝑥 − 3𝑦 = 10

Example 6 1.Mrs.Shubertsellsschool

suppliesinthemediacenter.Shesellserasersfor$2perpackageandpencilsfor$5perpackage.InAugust,shesold220packagesinallandmadeatotalof$695.Writeandsolveasystemofequationstofindthenumberofeachtypeofpackagesold.


Begin3-4Video3

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You Try: 1) Sugartopiasellssinglecrustapplepiesfor$6.99anddouble-crustedcherrypiesfor$10.99.ThetotalnumberofpiessoldonabusyFridaywasthirty-six.Iftheamountcollectedforallthepiesthatdaywas$331.64,howmanyofeachtypeofpiewassold?


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Practice 3-4 Solving Systems by Elimination Solve each system of equations by using the elimination method.

1. ) 2𝑥 − 4𝑦 = 10−2𝑥 + 6𝑦 = −4 2. ) 2𝑥 + 𝑦 = 3

−2𝑥 + 𝑦 = 1

3. ) 6𝑥 − 7𝑦 = −4−4𝑥 − 7𝑦 = 26 4. )𝑥 + 𝑦 = 30

𝑥 − 𝑦 = 6

5. )8𝑥 − 9𝑦 = 194𝑥 + 𝑦 = −7 6. ) 4𝑥 − 𝑦 = 6

3𝑥 + 2𝑦 = 21

7. )2𝑥 + 5𝑦 = −1𝑥 + 2𝑦 = 0 8. ) 2𝑥 + 5𝑦 = −10

−6𝑥 − 15𝑦 = 12

9. )−2𝑥 − 3𝑦 = −52𝑥 − 4𝑦 = −16 10. ) −10𝑥 + 6𝑦 = 0

−25𝑥 + 15𝑦 = 0



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Write a system of equations to model each scenario. Then solve the system. Make sure to give your answer in the context of the problem.

11.ApromoterattheHabershamCommunityTheaterpricedticketstoaplayasfollows:$17whenpurchasedinadvanceand$20whenpurchasedatthedoor.Thetotalnumberofticketspurchasedwas514,andticketsalestotaled$9,158.12.AtPet’sGoneWild,zebrafishcost$2.10eachandneontetrascost$1.85each,aftertaxes.IfMarshabought13fishforatotalcostof$25.80,aftertaxes,howmanyofeachtypeoffishdidshebuy?13.Alandscapingcompanyplacedtwoorderswithanursery.Thefirstorderwasfor13bushesand4trees,andtotaled$487.Thesecondorderwasfor6bushesand2trees,andtotaled$232.Thebillsdonotlisttheper-item.Whatwerethecostsofonebushandonetree?

Solve each of the following systems of linear equations with the method of your choice. Graphs are provided at the end of the section. 14. )6𝑥 + 3𝑦 = 27

𝑦 = 5 15. )𝑦 > −𝑥 − 3𝑦 ≤ 2𝑥 + 3

16. )4𝑥 + 6𝑦 = 3

𝑥 = 2𝑦 − 1 17. )8𝑥 − 6𝑦 = 18−𝑥 = 𝑦 − 4



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18. 9𝑦 ≥ <

;𝑥 − 3

𝑦 < @=;𝑥 + 2

19. ) 8𝑦 = −5𝑥 + 403𝑥 − 10𝑦 = −13

20. )𝑦 > 2𝑥 − 3

𝑦 ≤ 2𝑥 + 3 21. ) 4𝑥 − 3𝑦 = 113𝑥 − 5𝑦 = −11

22. J𝑦 = −2𝑥 + 2𝑦 = :

;𝑥 − 2 23. ) – 𝑦 = 1 + 2𝑥

3𝑦 = −6𝑥 − 3 Use these images for the inequality systems as well as any other systems that you choose to graph.



29



30

24. Write the equation of the inequality

25. What are the equations of the systems of inequalities represented

This WILL be on your

master

Spiral Practice Write your

Questions here!


31

26. Findtheslopeoftheline.Describehowonevariablechangesinrelationtotheother

a) ;theamountofwaterdecreasesby2

gallonsevery3minutes.

b) ;theamountofwaterdecreasesby2gallonsevery3minutes.

c) ;theamountofwaterdecreasesby3gallonsevery2minutes.

d) ;theamountofwaterdecreasesby1gallonperminute.

27.



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Lesson 3-5: Arithmetic Sequences Learning Target: I understand that sequences are a function with a domain of integers.

F.IF.3

A _______________ is an ordered list of numbers, pictures, letters, geometric figures, or just about any object you like. Each number, figure, or object is called a _______ in the sequence. For convenience, the terms of sequences are often separated by commas. We can also think of a sequence as a function whose inputs consist of ____________________________. With sequences, we use a new notation, called ______________________. In sequence notation, we refer the terms like this: a1, a2, a3,… a1 represents the _________ term in the sequence. a2 represents the ___________ term in the sequence. an represents the ____________ in the sequence. an-1 represents the _____________________ in a sequence. The subscript in each name is called the _________ and refers to the term number. In an arithmetic sequence, a common number (_______________________) is ___________to each term to get the next term in the sequence. Example 1: Are the sequences below arithmetic? If so, what is the common difference? 1A) 5, 10, 15, 20, 25…. 1B) 4, 8, 16, 32, 64….

Begin3-5Video1

Vocabulary: -Arithmetic Sequence -Continuous -Discrete -Explicit Formula -Recursive Formula



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You Try: Are the sequences below arithmetic? If so, what is the common difference? 1. 3, 9, 27, 81, 243 2. 3, 5, 7, 9, 11 3. 36, 30, 24, 18, 15….

We have two different ways to write formulas for sequences. The first method is called a _____________________. Recall that instead of using x and y or f(x) and x, we are going to use an and n in our formulas. Note: Both formulas are on your Algebra 1 Formula Sheet included on page 2 of your packet. The _________________ formula for an arithmetic sequence is as follows:

𝑎N = 𝑎N@= + 𝑑 where an = the current term an-1 = the previous term d = the common difference. Example 2: Write the recursive formula for the following sequence and then use it to find a7 3, 6, 9, 12, 15….. It is important to note that to be able use a recursive formula, you MUST have the _______________________.


Begin3-5Video2

Formula Sheet!


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You Try: Write the recursive formula for the following sequence and then use it to find a7 1.0,5,10,15,… 2.10,7,4,1,…

Using a recursive formula could be quite tedious for terms like the a25 or a100, as we would have to find every single term that came before them. Wouldn’t it be nice if we had another formula we could use that didn’t require the previous term? Thankfully, such a formula does indeed exist. The _______________formula for a sequence is as follows: 𝑎N = 𝑎= + (𝑛 − 1)𝑑 where an = the current term a1= the first term n = the index (a.k.a the term number) d= the common difference Example 3 Write the explicit formula for the following sequence and then use it to find a7 and a20. 3, 6, 9, 12, 15….. You Try: Write the explicit formula for the following sequence and then use it to find a7 and a20. 1.0,5,10,15,… 2.10,7,4,1,…

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Example 4 4a)If𝑎= = −9and

𝑎N = 𝑎N@= + 3Find𝑎;ExplicitorRecursive?____________

4b)If𝑎N = 7 + (𝑛 − 1)4Find𝑎:<

ExplicitorRecursive?________________

You Try: 1.If𝑎N = −5 + (𝑛 − 1)1 2.If𝑎= = 6and𝑎N = 𝑎N@= − 2Find𝑎:< Find𝑎;ExplicitorRecursive?____________________ExplicitorRecursive?____________________

Arithmetic sequences are __________ functions. However, since the inputs of sequences are restricted to the natural numbers, the graph of a sequence looks a little bit different. Instead of being one _____________ line, the graph ends up being _____________. This means that instead of all points being connected, we have to pick up our pencil to plot a set of points.

Begin3-5Video5

Write your Questions here! Begin3-5Video4


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Sometimes you may see sequences represented in graphs or tables.

Write the explicit and recursive formula for each example below.

Example 5 Example 6

You Try!

1. 2.



37

Example 7: The school cafeteria begins the day with a supply of 1000 chicken nuggets. Each student that passes through the lunch line is given 5 chicken nuggets. Write a numeric sequence to represent the total number of chicken nuggets remaining in the cafeteria’s supply after each of the first 6 students pass through the line. Include the number of chicken nuggets the cafeteria started with. Numeric Sequence for 1st 6 students: Recursive Definition: Explicit Definition: After how many students will there be only 15 nuggets left? You Try: Susana starts a job at Dairy Queen. She deposits $40 from each paycheck into her savings account. There was no money in the account prior to her first deposit. Write a numeric sequence to represent the amount of money in the savings account after Susana receives each of her first 6 paychecks. Numeric Sequence for 1st 6 paychecks: Recursive Definition: Explicit Definition: After how many paychecks will Susana have over $1000?


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Practice2-5ArithmeticSequencesForeachsequencebelow,findtherecursiveformula,explicitformula,andthegiventerm.1)–5,–3,–1,1,…. 2)3,9,15,21,…. 3)=

:, 1, ;

:, 2,…

Recursive:_____________ Recursive:____________Recursive:_____________Explicit:_______________ Explicit:_____________Explicit:_______________Find𝑎=::_______________ Find𝑎=>:_______________Find𝑎S:_______________4)5,3,1,–1,…. 5)4,8,12,16,…. 6)=

<, =:, ;<, 1,…

Recursive:_____________ Recursive:____________Recursive:_____________Explicit:_______________ Explicit:_____________Explicit:_______________Find𝑎S:_______________ Find𝑎=>>:_______________Find𝑎T:_______________Identifywhetherthefollowingdefinitionisexplicitorrecursiveandfindthegiventerm.7.If𝑎N = 8 + (𝑛 − 1)2 8.If𝑎= = −6and𝑎N = 𝑎N@= − 2Find𝑎:< Find𝑎;ExplicitorRecursive?____________________ExplicitorRecursive?____________________



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9.IfIf𝑎= = 3and𝑎N = 𝑎N@= + 7 10.𝑎N = 6 − 3𝑛

Find𝑎< Find𝑎=;ExplicitorRecursive?____________________ ExplicitorRecursive?____________________

Write the explicit and recursive formula for each example below. 11. 12.

13. Jacob uses 3 cups of flour in each cake he bakes. He starts the day with 50 cups of flour. Write a numeric sequence to represent the amount of flour remaining after each of the first 6 cakes Jacob bakes. Include the amount of flour Jacob started with. Numeric Sequence for 1st 6 cakes: Recursive Definition: Explicit Definition: WhatisthemostnumberofcakesthatJacobcanbake?

246810

121416

x y1 52 103 154 20



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14. The school cafeteria begins the day with a supply of 1500 tater tots. Each student that passes through the lunch line is given 7 tater tots. Write a numeric sequence to represent the total number of tater tots remaining in the cafeteria’s supply after each of the first 6 students pass through the line. Include the number of tater tots the cafeteria started with. Numeric Sequence for 1st 6 students: Recursive Definition: Explicit Definition: After how many students will there be only 100 tater tots left?

14.

Rewrite each of the equations into the slope

intercept form; y=mx+b. (Isolate the y).

15. 6x + 3y = 6 16. -2x + y = 8 17. 3x - y = 5 Graph the following equations. Be sure to put the equation in slope intercept form first. 18. 2x + y = 5 19. 3y = 2x - 3

This WILL be on your mastery check!

Spiral Practice


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20. Write the explicit and recursive formula for each example below. Then make a table and graph

n an



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Unit 2 Study Guide 1. Describe how you know that a system of equations has one solution:

Graphically:__________________________________________ Algebraically:________________________________________ 2. Describe how you know that a system of equations has no solutions: Graphically:__________________________________________ Algebraically:________________________________________ 3. Describe how you know that a system of equations has infinitely many solutions: Graphically:__________________________________________ Algebraically:________________________________________ 4. Which ordered pair is a solution to the following system?

)−3𝑥 − 4𝑦 = −255𝑥 − 12𝑦 = 23

A) (1, –1) B) (7, –1) C) (7,1) D)(1,1) 5) To solve the following system by elimination, what would you multiply the first equation by if you were trying to cancel the y values?

) 2𝑥 − 2𝑦 = 2−3𝑥 + 4𝑦 = 1

6) You are selling tickets at a high school football game. Student tickets cost $3

and general admission tickets cost $5. You sell 1115 tickets and collect $3955. How many of each type of ticket did you sell?


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7) Write the system of inequalities that is represented by the graph below?

Name two ordered pairs that are solutions:_________________ Name two ordered pairs that are not solutions: ____________________ 8.a)Anonlinedigitalmovierentalcompanycharges$5.95forthefirstdigitmovieand$3.95foreachadditionalmovie?Writetheexplicitformulaforthescenario.Numberofdigitalmovies

1 2 3 4 5

Cost 5.95 9.90 13.85 17.80 21.75b)Usingtheabovescenario,whatwoulditcosttowatch20digitalmovies.c)Writetherecursiveformulaforthesequence.9.Writetheexplicitandrecursiveformulaforthegraphtotheright:

246810

121416


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10.Giventhefollowingsequence7,12,17,22,27,…Writetheexplicitandrecursiveformula.11) Graph the system and tell how many solutions. If there is one solution, list the ordered pair that is the solution.

y = 2x – 5 x + y = 7

How many solutions? ______ If one, what is it? _________

12) Graph the system and tell how many solutions. If there is one solution, list the ordered pair that is the solution. y – 2 = x y = x +5 How many solutions? ______ If one, what is it? _________

13) Graph the system and tell how many solutions. If there is one solution, list the ordered pair that is the solution. 3x + 3y = 3 y = -x + 1 How many solutions? ______ If one, what is it? _________


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14) y = 3x y = 4 – x 16) 2x + y = 4

x – 2 = y 18) 2x – 2y = 2

–3x + 4y = 1 20) x – 4y = 24 3x – 4y = 16.

15) -3x – 3y = -8 -6x – 6y = -12 17) 5x + 3y = 3 3x + 4y = 15 19) x = 4 2x + 3y = 5 21) 5x + y = 14 -10x – 2y = -28


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21) Graph the system of inequalities.

22) Graph the system of inequalities y > 0 x> 3

23) Yaneli found a pre-paid phone plan that charges 15 cents per minute for phone calls and 25 cents per text message sent. a) Write an equation to model the cost (C) of talking minutes (M) and sending texts (T). b) How much will it cost Yaneli to send 20 text messages and talk for 55 minutes? c) If Yaneli knows he will send 20 text messages and only has $15, how many minutes can she talk? 25) Car A cost $10,000 to buy and cost an average of $0.50 per mile to maintain. Car B cost $18,000 to buy and costs an average of $0.25 per mile to maintain. After how many miles would the total cost for each car be the same?

€

y ≥ −x+1

y ≤23x+ 2


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26) Lea and Julie are selling pies for a school fundraiser. Customers can buy cherry pies and blackberry pies. Lea sold 9 cherry pies and 5 blackberry pies for a total of $196. Julie sold 14 cherry pies and 13 blackberry pies for a total of $378. Find the cost each of one cherry pie and one blackberry pie? 27) At a gym, a group of 3 members and 4 nonmembers pay a total $125. Another group of 1 member and 5 nonmembers total $115. Based on this information of the separate groups, how much does it cost members and nonmembers at the gym. 28) At the football game, a bag of popcorn cost $1.50 and a hamburger cost $3.50. After the game, the concession stands had sold a total of 100 items (hamburgers and bags of popcorn). They made $230 from selling hamburgers and bags of popcorn. How many of each item were sold? 29)Usethegraphatthelefttographthefunctionsthatallowyoutofindthesolution(s)totheequations. 2𝑥 + 5 = 𝑥 f(x)=_________________g(x)=__________________Solution:__________________________


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22) Jason is buying wings and hot dogs for a party. One package of wings costs $8. Hot dogs cost $5 per package. He must spend less than $40. Jason knows he will be buying at least 4 packages of hot dogs. Write a system of inequalities to model the situation. Graph both inequalities and shade the intersection.

A) Write a system of linear inequalities.

B) Graph the inequalities

c) Could Jason buy 3 packages of wings and 5 packages of hotdogs? Explain why or why not?

Begin reviewing for your test!


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Glossary

Arithmetic Sequence: A sequence of numbers in which the difference between any two consecutive terms is the same Continuous: Describes a connected set of numbers, such as an interval Discrete: A set with elements that are disconnected Explicit Formula: A formula that allows direct computation of any term for a sequence a1, a2, a3,…,an,…. Recursive Formula: A formula that requires the computation of all previous terms to find the value of an. Substitution: To replace one element of a mathematical equation or expression with another.

Documents

Algebra 1 Unit 3 Systems of Linear Equations and ... · Unit 3: Systems of Equations and Inequalities 16 Lesson 3-3: Solving Systems by Substitution Learning Target: I can solve systems