1. The lines intersect. There is one solution, the point where they intersect. The system is called a consistent system.

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1 Commo Core Math 3 Notes Uit Day Systems I. Systems of Liear Equatios A system of two liear equatios i two variables is two equatios cosidered together. To solve a system is to fid all the ordered pairs that satisfy both equatios. Whe solvig a system three situatios ca occur:. The lies itersect. There is oe solutio, the poit where they itersect. The system is called a cosistet system. 2. The lies are parallel. There is o solutio, the lies do ot itersect. The system is called a icosistet system. 3. The lies are the same. There are ifiitely may solutios, the lies coicide. The system is called depedet. Systems of equatios ca be solved by various methods which will be discussed i this chapter. Solve each system by graphig. A. x2y0 y2x3 B. 2 x y 4 xy2 Math 3 Hoors Notes Uit Modelig with Liear Fuctios

2 II. Word Problems. Travis ad his bad are plaig to record their first CD. The iitial start-up cost is $500 ad each CD will cost $4 to produce. They pla to sell their CD s for $0 each. How may CD s must the bad sell before they make a profit? 2. A service club is sellig copies of their holiday cookbook to raise fuds for a project. The priter s setup charge is $200 ad each book costs $2 to prit. The cookbooks will sell for $6 each. How may cookbooks must the members sell before they make a profit? III. Substitutio Method To solve a system of equatios by the substitutio method:. Solve oe of the equatios for oe of the variables.(choose a variable with a coefficiet of or - if possible.) 2. Substitute this expressio ito the other equatio to produce a equatio with oly oe variable. 3. Solve the equatio i Step 2 for the remaiig variable. 4. Substitute this solutio ito the expressio obtaied i Step. 5. Solve for the secod variable. 6. Write your solutio set as a ordered pair, ad check i each equatio. Solve each system by substitutio. A. 2x y 8 3x y 2 B. 5 x 3 y 2 x2y3 Math 3 Hoors Notes Uit Modelig with Liear Fuctios 2

3 IV. Elimiatio Method Steps for elimiatio-by-additio method:. Write each equatio i stadard form if eeded. 2. If ecessary, multiply oe or both equatios by some costat which will make the x or y coefficiets opposites. 3. Add the equatios from step 2 together elimiatig oe of the variables. 4. Solve for the remaiig variable. 5. Substitute this solutio ito either of the origial equatios. 6. Solve for the secod variable. 7. Write the solutio set ad check. Solve each system by elimiatio. A. 3 x 5 y 6 2x 7y 4 B. 6 x y 5 4x 3y 7 C. 9 x 2 y 3 3x 4y 2 Math 3 Hoors Notes Uit Modelig with Liear Fuctios 3

4 V. Solve by Graphig. Review: Graph by usig x ad y itercepts. To fid the x-itercept: To fid the y-itercept: Substitute a 0 for y ad solve for x. Substitute a 0 for the x ad solve for y. (x, 0) (0, y). Graph usig x ad y itercepts. A. 2x4y 8 B. 3 2 x y D. Sketch the graph. Fid the coordiates of C. Solve by graphig. the figure formed or fid the coordiates of the feasible regio x3y6 x 3 3x2y8 y 3 y 2 yx Math 3 Hoors Notes Uit Modelig with Liear Fuctios 4

5 Commo Core Math 3 Notes - Uit Day 2 Liear Programmig Problem : A calculator compay produces a scietific calculator ad a graphig calculator. Log-term projectios idicate a expected demad of at least 00 scietific ad 80 graphig calculators each day. Because of limitatios o productio capacity, o more tha 200 scietific ad 70 graphig calculators ca be made daily. To satisfy a shippig cotract, a total of at least 200 calculators must be shipped each day. If each scietific calculator sold result i a $2 loss, but each graphig calculator produces a $5 profit, how may of each type should be made daily to maximize profits? X: umber of scietific calculators Y: umber of graphig calculators produced. What do the followig costraits mea? a. x00 b. y 80 c. x 200 d. y 70 e. x y The above costraits are graphed below. Oe of the vertices of the feasible regio is (20,80). Name the rest of the vertices ad idicate what each oe meas. (There are 5 total) a. b. c. d. e. Math 3 Hoors Notes Uit Modelig with Liear Fuctios 5

6 3. Each scietific calculator sold results i a $2 loss, but each graphig calculator produces a $5 profit. The equatio P 2x 5y represets this situatio. Explai each part of the equatio. a. P represets? b. -2x represets? c. 5y represets? d. If 50 scietific calculators were sold ad 80 graphig calculators were sold, what would be the profit? 4. Usig the profit equatio ad the vertices you foud i step 2, fid the profit for each vertex ad explai what that meas. a. b. c. d. e. 5. How may of each calculator must be sold to geerate the largest profit? What is the largest profit? Problem 2: A backpack maufacturer produces a iteral frame pack ad a exteral frame pack. Let x represet the umber of iteral frame packs produced i oe hour ad let y represet the umber of exteral frame packs produced i oe hour. The iequalities describe the costraits for maufacturig both packs. x3y8 2x y6 x 0 y 0 Graph the costraits (o the ext page) Math 3 Hoors Notes Uit Modelig with Liear Fuctios 6

7 . Graph the costraits. (o the ext page) 2. Name the vertices of the feasible regio (there are 4). 3. Fid the equatio that represets the profit if each iteral frame pack profits $50 ad each exteral frame pack produces a profit of $ Usig the vertices ad the profit fuctio, determie the maximum profit ad the umber of each type of backpack eeded to produce. Summarize Steps i Liear Programmig: Math 3 Hoors Notes Uit Modelig with Liear Fuctios 7

8 Problem 3: Suppose you make ad sell ski lotio. A quart of regular ski lotio used 2 cups of oil ad cup of cocoa butter. A quart of extra-rich ski lotio cotais cup of oil ad 2 cups of cocoa butter. You make a profit of $0 per quart o regular lotio ad a profit of $8 per quart o extra-rich lotio. You have 24 cups of oil ad 8 cups of cocoa butter. How may quarts of each lotio should you make to maximize your profit? What is the maximum profit? Math 3 Hoors Notes Uit Modelig with Liear Fuctios 8

9 Problem 4: Two raw materials are eeded to make oe of the products produced by Dartmouth Ic. The product must cotai o more tha 9 uits of material A ad at least 8 uits of material B. The compay ca sped o more tha $300 o materials for each piece produced. Material A costs $4 per uit ad weighs 0 pouds per uit. Material B costs $2 per uit ad weighs 20 pouds per uit. How much of each material should be used to maximize the weight of the product? Math 3 Hoors Notes Uit Modelig with Liear Fuctios 9

10 Problem 5: Jerry speds o more tha 20 hours a week workig at two jobs durig the school year. He is paid $0 a hour for tutorig Algebra 2 studets ad $7 a hour for deliverig pizzas for Pizza Hut. He wats to sped at least 3 hours, but o more tha 8 hours a week tutorig. Fid Jerry s maximum weekly earig Math 3 Hoors Notes Uit Modelig with Liear Fuctios 0

11 Commo Core Math 3 Notes - Uit Day 3 Arithmetic Sequeces ad Series I. Itroductio Towerig Numbers Row # of Bricks Complete the table to the right of the tower.. There are six rows i the tower pictured above. How may bricks would be i the seveth row? 2. Suppose you wated to build a tower with 25 rows usig the same desig. Describe how you could figure out how may bricks you would eed for the twety-fifth row. 3. If somebody told you how may rows of bricks were i a tower, how could you figure out the umber of bricks i the logest row? 4. A very large tower was build usig the same desig. The logest row had 299 bricks i it. How may rows of bricks did the tower have? 5. If somebody told you how may bricks were i the logest row of a tower, how could you figure out how may rows there were? Math 3 Hoors Notes Uit Modelig with Liear Fuctios

12 II. Arithmetic Sequeces Defiitio A sequece i which each term after the first is foud by ADDING a costat to the previous term called the Commo Differece. Formula: Examples:. Fid the first five terms if a 5 ad d What term is -3,, 5, 9 29? is the th term of 4, 2.5,, Fid the idicated term: a 5 for 3,3, Write a equatio for the th term of the sequece: 8, 7, 26, Fid the missig terms: 3,,, Fid the missig terms:, 4,,,, 29 Math 3 Hoors Notes Uit Modelig with Liear Fuctios 2

13 III. Arithmetic Series Defiitio: A sequece writte as a sum. Formula: Examples:. Fid the sum of 5, 7, 9,, Fid the sum of the first 50 terms where a 5 ad d Name the first three terms of the series where a 4, a = -85 ad S Name the first three terms of the series where a 5, a = 00 ad S 050 IV. Sigma Notatio Summatio Notatio Types ad their formulas Costat Liear Quadratic Examples: Math 3 Hoors Notes Uit Modelig with Liear Fuctios 3

14 Expad the series ad fid the sum.. 0 k2 2k Expad the series: Method : Method 2: x4 3x 6 Fid Fid a Fid a Fid the sum a5 3a 4 Math 3 Hoors Notes Uit Modelig with Liear Fuctios 4

15 Commo Core Math 3 Notes - Uit Day 4 Geometric Sequeces ad Series I. Itroductio. Nigel saves $2 i the first week of the New Year. If he doubles the amout he saves every week after that, how much will he save i the 2 d, 3 rd, 4 th, ad 5 th week of the year? Week # st Week 2 d Week 3 rd Week 4 th Week 5 th Week Amout Saved 2. The price of a item triples o the first day of every moth. It costs $2.00 o the st of Jauary. Calculate its cost o the first day of the followig four moths. Date st February st March st April st May Cost 3. A tow had a populatio of 00. I Jauary 200, a ew factory was beig built i the tow ad cosequetly, the local authority expected that the tow s populatio would icrease by 20% year-oyear for the et 5 years. a. Explai clearly how the tow s populatio i Jauary 20 is calculated. Date Ja 20 Ja 202 Ja 203 Ja 204 Ja 205 Populatio 4. Examie the completed tables from the previous three questios. a. Do the umbers you calculated follow ay particular patter i each table? Table? Table 2? Table 3? Math 3 Hoors Notes Uit Modelig with Liear Fuctios 5

16 II. Geometric Sequeces Defiitio: A sequece i which each term after the first is foud by multiplyig a costat called a Commo Ratio. Formula: To fid r:. Fid the seveth term of a geometric sequece i which a3 96 ad r Fid the th term where a4 0, 0 ad r Fid the th term where a6 5, a7=5 ad 2 4. Write a equatio for the th term of the geometric sequece: 3, 2, 48, 92, 5. Fid the missig geometric meas:,, 64,,,,, 2, 6. Fid the missig terms:, 3,,,, 48 Math 3 Hoors Notes Uit Modelig with Liear Fuctios 6

17 III. Fiite Geometric Series Formula: A sequece that is writte as a sum. Examples:. Fid the sum where a 2, a5 972, ad r 3 2. Fid the sum where a 5, r 3, ad 7 3. Fid the sum of = Fid the sum: Fid the sum: x3 23 x 6. Fid a if S 39,360, 4 ad r 3 Math 3 Hoors Notes Uit Modelig with Liear Fuctios 7

18 IV. Ifiite Geometric Series How ca somethig ifiite have a sum? Formula: where r. Examples:. Fid the sum of Fid the sum of 4/3-2/3 + /3 /6 3. 3(2) a 4. a a (4) 2 a Math 3 Hoors Notes Uit Modelig with Liear Fuctios 8

19 Commo Core Math 3 Notes - Uit Day 5 Arithmetic ad Geometric Sequeces ad Series Applicatio I. Itroductio To Babysit or Not To Babysit? This summer, Shara is lookig to babysit to make some extra moey. She has the choice betwee babysittig for two families. The Pi Family offered Shara a flat $00 stiped for gas moey plus $75 a day to babysit. The Radical Family had a differet approach. Sice they wated to esure Shara would stay with them for the etire moth, they offered to oly pay her wheever she stopped workig for them i oe lump sum. She would be paid a iitial amout of a pey for choosig their family ad the her pay would double each day she babysat util she decided to stop babysittig for them or whe the moth eded. (For example, she would ear $0.0 iitially, $0.02 for the oe day, $0.04 for two days, ad so o. This meas that the startig pay is after oe day. ) If at ay time, she wated to stop babysittig, they would give her the moey she eared up util that poit. Which family should Shara work for, assumig she would babysit o weekdays oly?. What fuctio describes the Pi Family s offer? 2. What fuctio describes the Radical Family s offer? 3. If Shara could oly babysit for the first three weeks, would this chage your choice? Why or why ot? Justify your aswer usig good mathematics. 4. Is there a poit at which both families would pay her the same amout? 5. The Radical Family asked Shara to babysit four additioal days that same summer ad cotiue o her pay system. If she worked the three weeks ad four additioal days, would that chage your mid? Math 3 Hoors Notes Uit Modelig with Liear Fuctios 9

20 II. Idetify whether the sequece is arithmetic, geometric, or either. If the sequece is arithmetic or geometric idetify the commo differece or commo ratio.. 40, 43, 46, 49, ,2, 36,08, ,6,36,64, a a a (2 ) 2 4( 3) III. Applicatio Geometric Formulas: Use if give the iitial amout. Use if give the first term. (Amout at = ) a a r 0 a a r. Edgar is gettig better at math. O his first quiz he scored 57 poits, the he scores 6 ad 65 o his ext two quizzes. If his scores cotiued to icrease at the same rate, what will be his score o his 9 th quiz? Show all work. 2. A recoverig heart attack patiet is told to get o a regular walkig program. The patiet is told to walk a distace of 5 km the first week, 8 km the secod week, km the third week ad so o for a period of 0 weeks. At that poit the patiet is to maitai the distace walked durig the 0 th week. How far will the patiet walk durig the 0 th week? 3. Suppose you drop a teis ball from a height of 5 feet. After the ball hits the floor, it rebouds to 85% of its previous height. How high will the ball reboud after its third bouce? Roud to the earest teth. Math 3 Hoors Notes Uit Modelig with Liear Fuctios 20

21 4. A virus reproduces by dividig ito two, ad after a certai growth period, it divides ito two agai. As the virus cotiues to reproduce, it will cotiue to divide i two. How may viruses will be i a system startig with a sigle virus AFTER 0 divisios? 5. A house worth $350,000 whe purchased was worth $335,000 after the first year ad $320,000 after the secod year. If the ecoomy does ot pick up ad this tred cotiues, what will be the value of the house after 6 years? 6. Viola makes gift baskets for Valetie s Day. She has 3 baskets left over from last year, ad she plas to make 2 more each day. If there are 5 work days util the day she begis to sell the baskets, how may baskets will she have to sell? 7. Sam has purchased a $30,000 car for his busiess. The car depreciates 30% every year. Depreciatio meas the value of the car goes dow by that percet each year. What will be the value of the car after the 5 th year? Note: The car is 0 years old whe purchased so the first year is the secod etry i the sequece. 8. Alle is o the football team this year but he has poor time maagemet skills. His mother told him that he is off the team if he fails aythig i school. O his first math quiz he eared a 90, the he eared a 86 ad a 82 o his ext two quizzes. If his grades cotiue at this rate, what will his quiz grade be after the 8 th quiz? Will he still be o the team? 9. I a certai regio, the umber of highway accidets icreased by 20% over a four year period. How may accidets were there i 2006 if there were 520 i 2002? Hit: Whe the percet icreases, you wat the origial 00% plus the additioal 20%. Math 3 Hoors Notes Uit Modelig with Liear Fuctios 2

22 Commo Core Math 3 Notes Uit Day 6 Recursive ad Explicit Formulas I. Itroductio Math 3 Hoors Notes Uit Modelig with Liear Fuctios 22

23 II. Special Sequeces,, 2, 3, 5, 8 Ca you fid the ext term? Ca you fid the 30 th term without havig to fill i the blaks? This sequece is a famous sequece called. Term First Term Secod Term Third Term Nth Term Represeted by The patter is a example of a which meas that Example : Fid the first five terms of the sequece i which a 5 ad a 2 a+7 ad. = = 2 = 3 = 4 First five terms: Example 2: Fid the first five terms of the sequece i which a 3 ; a a ad Math 3 Hoors Notes Uit Modelig with Liear Fuctios 23

24 III. Arithmetic Explicit ad Recursive Formulas Explicit Formula: Recursive Formula: a a ( ) d a start a a d Hopefully you are remided of: Now = start Next = ow + d Example : 7, 0, 3, 6, Explicit Formula: Recursive Formula: Example 2: 386, 365, 344, 323, Explicit Formula: Recursive Formula: IV. Geometric Explicit ad Recursive Formulas Explicit Formula: Recursive Formula: a ar a start a ar Hopefully you are remided of: Now = start Next = (ow)(r) Example : 297, 99, 33,, Explicit Formula: Recursive Formula: Example 2:, -6, 36, -26, Explicit Formula: Recursive Formula: Math 3 Hoors Notes Uit Modelig with Liear Fuctios 24

25 Determie whether the sequece is arithmetic or geometric. The write the explicit ad recursive formula.. 40, 43, 46, 49 Explicit Recursive 2. 3, 2, 48, 92 Explicit Recursive , -26, -8, -0 Explicit Recursive , 8, 59, 29.5 Explicit Recursive Determie whether the formula is arithmetic or geometric ad explicit or recursive. The write the opposite formula for each sequece.. a ( 3) 2. a a a a a a a 43 Math 3 Hoors Notes Uit Modelig with Liear Fuctios 25