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Transcription

1 31 Systems of Equations System: or sinayamousae#eegiens What could these two lines look like? at theorem tee?± net?eeeaaakee:eyitgeers What does it mean to be a solution of a system? That solution make Ex: Solve graphically stroll the eyes { Ente try L * j IIIxn z I + y Ex E

2 yl LcD8 tn/8dl8tdxgy82x2x+8y3zx5zy 2 +8 f " # hy E i D39Eo

3 q 3 yie " ait?i# :/ yl

4 { Ex: Solve graphically zy 2 44 AO 4 6 Zx y 4 C D Zx G) y G) f 4) Zx + y 4 [ tzxytt#, I 6+410

5 BO E y 2 3 Ff *,

6 28 o f of * Parallel 2 44 Lines It No solution m2 K 42 3 M2 < ' in,:# ii" ( 0,4 ) G, ) L

7 32 Substitution & Elimination Algebraic (rather than graphing) solutions to systems of equations Substitution Shoving Ex: Solve and Check 1) 2) { 3y+x4 3 ytlzy 1 BO is isolated D 4 substitution 3y+2y 14 ' BO X ZY I 11

8 XH, y I or ( I,1) Check : AO 3y+x 4 3C c) + I I 4 3yµ BO Zy 1 I 21 XI I 1 '

9 2 { * ±s 01*02 Oly }x tgy 3 5,9 },EFyF } olyx 0 Oly } 2 2 y 2 y 10 ( Oly ) x is isolated AO 2 46 ZG y) zy 65 4E?Y 46 4 Y

10 44, Endx : zc Z Y Y f I + E Et k} oe±e ofd I }f I±E+ check : DO 0 lx0 to ly OA F*Ey Fox to toy } 's ' 2 + CEE E±ot±ov

11 Substitution 1 Isolate one variable ( Solve for x or y) 2 Substitute this into the other eqtt 3 Solve ( Shoving turducken ) 4 Solve for the other 5 Check Variable

12 Ex : Solve by substitution { y 1 1 Solve for one Variable y 3 +5 y 3 5

13 @ Substitute into the other eqtt : y 1 X + (3 5)1 3 X Solve for y y3x 5 y3t 5 y 2 l, y 2

14 5 Check C2) X + YII 1 + C 2) I

15 Elimination Adding First Don t look at this Let a b c edgedd Then a + c b + c Since a + * b tr and a + c b + d Since c d So a b c c nard d a + c b + d Ok, look back now

16 Ex: Solve and Check 1) { Iihf Find y : 3 y 3 ofz9 ± E 2 (3) + ( o ) E 6 GOD )

17 2) {?I ) y 6 : Kndx I IYI tgf check : B 4)

18 3) { 10 a a No elimination in 10 at 6b 6 4 a + 3b z 10 a + 6b 6 8A 6b F f 0 Za a 0 6 5) b : that 6b 6 ' on + ;@ Tat bag 's :b 3 4 ( o ) +34 ) I

19 4) { 4 3 ( x 3 G + y ) b) 2 y 6) y Y Yx y Zy y tn et EI 9 90 by ; 7) Y Y 6 y

20 Check both : 6, ( zty ) 4 6 I 3 (2+6) 24 3 ( 8) 3 ( x 3 ( 6 lot 3 t 4) Zy io ) I 2 ( 6 ) 12 Escoffier 36 Problem Solving Using Systems Two Equations 1) A total $10000 was invested in two accounts for one year If the first account earned 5% and the second account ee earned 25%, and the investments earned a total of $1500 together, how much was invested in each account?

21 5 { 5 9*7 7y 3x a , y 4 y 4g I Fndx : sxat#jiitfey 45

22 x y I I or y 05 X 05 Check : 7y 3 5 TC 05) 3105 ) I 5 5

23 10a + 6b 8 6) { 5a + 3b 2 loat6b na 6b 4 No Inconsistent Parallel False Solution System Lines 7) { 2s 13t s + 91t 840 ±, lt 840 t + 00 True 840 tfdspmueenttisyesiem ( Infinite # Solutions

24 36 Problem Solving Using Systems of Two Equations 1) A total of $10000 was invested in two accounts for one year If the first ) account earned 5% and the second account earned 25%, and the investments earned a total of $1500 together, how much was invested in each account? + tempted mn#ioy x+yia oo 0EI Ioo 2) I want to obtain 160 centiliters of a 70% alcohol solution by mixing distilled water with an 80% alcohol solution How much of each should I use? Mee OF Mint#aotawhd t M I E #6oa o x+ Ypz fegfeoh FI#gd?l4@fy:z* [ 70% Pureakohol

25 1 +y 10, , , y 100,000 y 5,000 Find : ,000 X 5,000 Tto#sfsooD+o25CsooDE FIFE

26 3) A chemical stockroom has a 20% alcohol solution and a 50 % solution How many deciliters of each should be used to obtain 90 deciliters of a 30% solution? Mix tore 0 i IE EI I?F *ii* Ig : I o*± ) A tea shop manager wants to blend tea that sells for $5 per kilogram with tea that sells for $650 per kilogram to produce a blend that will sell for $6 per kilogram How much of each should be used to obtain 75 kilograms of the new blend? ftpmsnes#ieyyt 6 ( 75 ) 450 t + 75 y 5t + 6 5y yso 5y It; t y tfs 375 yeg te#

27 5) A gourmet food store manager wishes to blend coffee that sells for $7 per kilogram with coffee that sells for $ 950 per kilogram to produce a blend that will sell for $8 per kilogram How much of each should be used to obtain 100 kilograms of the new blend? 6) A farmer wants to divide up her square field into three sections, for her cows, her goats, and her chickens What are the dimensions of her field if she ll need 120 meters of fence to complete her project? x Total fence lzom 20in # X X ZOM or20 ZOM by 20 Dimensions : 6 120am or length width 2 On 2om

28 7) That same farmer realizes that she mismeasured her field, and it s not actually a square In fact, the width is twice as long as the length, and, in the time it took her to remeasure the field, all her chickens flew away If she decides to divide up the field into two equal sections, with the dividing fence running parallel to the longer side of the field, what are the dimensions of the field? II 1203 the width is twice as w Zl w + zl long as the length ) 8) You want to hire a magician for a friend s party There are two magicians available on the day of the party the first magician charges a $25 flat fee, plus $750 per hour, including travel, while the second magician charges a $37 flat fee, plus $6 per hour, including travel What s the breakeven point for this situation? Using the breakeven point, decide which magician should be hired if the party plus travel will take 6 hours Magl : Fee : $25 + $7501 hour Mag 2 : Fee : $37 + $61 hour p Variable Fixed ( Doesnuntonge ) T h # hours # I : Cost h # 2 Cost : 37T 6h

29 7 W Zl 120 3W + 2 l Substitution 120 3W + 2 l 1203 ( zl ) + 2 l lt 2 l l 1 l ) l 15 W l 30 length width 15 m 30 m

30 5h 8 Cost # h Cost # h Break even : The costs are the same h 6h 6h h h 12 I Ts h 8 hours < 6 hours 8 hours 1st Magician Is cheaper

31 Word Problems Answers: 1) $5,000 was invested in each account 2) You should use 20 cl of distilled water and 140 cl of the 80% solution 3) 60 dl of the 20 % solution and 30 dl of the 50 % solution should be used 4) You should blend 25 kg of the $5 tea with 50 kg of the $650 tea 5) You should mix 60 kg of the $7 coffee with 40 kg of the $950 coffee 6) Her field is 20 m x 20 m 7) Her field is 30 m x 15 m 8) The break even point happens at 8 hours You should hire the first magician