Systems of Equations and Inequalities

Transcription

1 Chapter 6 Sstems of Equations and Inequalities Prerequisite Skills for the chapter Sstems of Equations and Inequalities. The least common multiple of 0 and is 0.. Two lines in the same plane are parallel if the do not intersect } } The lines are not parallel because their slopes are not the same.. m > 9 m > 9. 6t t < < > Lesson 6. Graphing Solve Linear Sstems b Investigating Algebra Activit for the lesson Solve Linear Sstems b Graphing Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 7. m m (z ) z 6 m 0 0z 0 z 6 9. m 6 z 0 6 m z z The lines are not parallel because their slopes are not the same. 0. The lines are parallel because their slopes are both The lines are parallel because their slopes are both The solution is.. Bill and his brother will have the same number of books in months. The will each have books.. (, ) 7 The graphs intersect at the point with the -value that is the solution to the two equations. Algebra Worked-Out Solution Ke 8

2 } } (, ) The solution is, (, ) () 0 () 0 Because the ordered pair (, ) is a solution of each equation, it is a solution of the sstem.. The solution is, (, ) (, ) () 0 The solution is,. Lesson 6. Solve Linear Sstems b Graphing Guided Practice for the lesson Solve Linear Sstems b Graphing. () () 0 Because the ordered pair (, ) is a solution of each equation, it is a solution of the sstem (0, 0) (, ) 0 0 (, ) 0 () () Because the ordered pair (, ) is a solution of each equation, it is a solution of the sstem. 86 Algebra Worked-Out Solution Ke 0 0 (0) (0) The cost is the same at 0 sessions. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved..

3 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. 0 (, 9) 9 0 () 9 0 () The cost is the same at sessions } 0 (, 8) () 0(8) biccles were rented. Eercises for the lesson Solve Linear Sstems b Graphing Skill Practice. A solution of a sstem of linear equations in two variables is an ordered pair that satisfies each equation in the sstem.. Graph both equations on the same coordinate plane. Find the point where the lines appear to intersect, substitute the and values from this point into each equation. If the ordered pair is a solution to each equation, it is a solution of the sstem.. 0 () 0 (, ) is a solution.. 8 () () 0 () 8() 0 6 Þ (, ) is not a solution () () 0 7 () Þ 7 Þ 0 (, ) is not a solution. 6. B; (0, ) () (0) 0 8 Þ 8 0 () 0 7(0) () B; (, 6) () () 0 0() () 0 Þ Þ () (6) 0 0() (6) 0 8. (, ) () 0 () () 0 (, ) is a solution. 9. (, ) 6 0 () (, ) is a solution. 0. (, ) 0 () 0 (, ) is a solution.. The error is that the -intercept of equation should be, not. When graphed correctl, the lines intersect at (, ). 6 () 0 6 () () (, ) is a solution to the linear sstem.. Test (, ). (, ) 0 0 (, ) is a solution. Algebra Worked-Out Solution Ke 87

4 . 8 (, 0) 6. } } 8 Test (, 0) () (, 0) is a solution (, ) Test (, ). 0 () 0 () 6 (, ) is a solution (, ) Test (, ). () 0 () (, ) is a solution. Test }, }. (, ) } } 0 } } 0 }, } is a solution. 7. (0, ) Test (0, ). (0) () 0 (0, ) is a solution } } 6 Test (, ). (, ) } () () 0 () () (, ) is a solution. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 88 Algebra Worked-Out Solution Ke

5 } (7, ) 8 (, ) Test (7, ). 6(7) () 0 7 (7) () (7, ) is a solution } 7 } Test (, ). () 0 9 () () (, ) is a solution } } 0 9 (, ) (, 6) 0 7 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Test (, ). 7() () 0 9() () 0 (, ) is a solution (, ) 0 Test (, ). } } } 6() () 0 6 () () (, ) is a solution. Test (, 6). () (6) 0 () (6) (, 6) is a solution.. } } } } } } Test (8, ). } } } } 6 } 8 } (8, ) } (8) } () 0 } 8 } () 0 } } } } } (8, ) is a solution. Algebra Worked-Out Solution Ke 89

6 . } } } 8 } 8 } 8 } (, 6) 8 9. a. } 6 } 6 } } b. } 6 } 6 (, ) Test (, 6). } () } (6) 0 } 8 } () 6 0 } 8 } 8 (, 6) is a solution Test (, ) (, ).6().() 0.6().6() 0 6 (, ) is a solution. 7. Answers will var Check (0., 0.). (0., 0.) (0.) (0.) It is important to check our solution because the coordinates of the solution ma not be obvious b looking at the graph. Test (, ). 0 } () 6 0 } () The solution is (, ). c. The two equations from the sstem in part (b) were set equal to each other for the equation in part (a). d. Set each side of the equation equal to. } } Graph both equations on the same coordinate plane. The point where the graphs intersect is the solution. The -coordinate of the intersection point is the value of in the given equation } } 9 } } 9 Line & : Test (, ). (, ) (, ) () () 0 () 0 Line & : Test (, ). (, ) () 0 () Line & : Test (, ). () 0 9 () () The vertices of the triangles are (, ), (, ), (, ). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 90 Algebra Worked-Out Solution Ke

7 Problem Solving b. Let minutes on elliptical trainer. Let minutes on stair machine.. The percent who watch hour or less will equal the percent who watch more than hour 0 ears after 990, or in B; 8 7, 8 8 } 0,80. Let number of small cards. 8 Let number of large cards (, ) } 9 You should spend minutes on the elliptical trainer and minutes on the stair machine.. a. 8 b. (, 0) 9 Test (, 0) 0 0 () (0) She sold small cards and 0 large cards. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. a. Let minutes on stair machine c. Let minutes on stationar bike } (, ) 6 You should spend minutes on the stair machine and minutes on the stationar bike. It makes sense to become a club member if ou attend more than movies a ear. The graph shows that the -values (total cost) are higher on the line representing a non-member when (number of movies viewed) is greater than. 6. Let purchase price. Let amount paid (, 00) 0 You should choose $ off if our purchase is less than $, and ou should choose 0% off if our purchase is more than $, because 0% of amounts greater than $ is more than $. Algebra Worked-Out Solution Ke 9

8 Graphing Calculator Activit for the lesson Solve Linear Sstems b Graphing Check: 0 () 0. Y 0 () 0 6 Y. 7 The solution is about (.,.) Y ( 7) 0 () Y The solution is about (, ). () The solution is (, ). Y 0.. Check:.8.8 () () 0 7 Y (0) Y 0. 0 The cost for the internet service provider is $9 after 0 months (0) Y. Lesson 6. Solve Linear Sstems b Substitution The cost for the website hosting compan is also $9 after 0 months...9 Guided Practice for the lesson Solve Linear Sstems b Substitution The total cost will be the same for both companies after 0 months (6) () The solution is (, 7). Check: 7 0 () () m The solution is (0, ). Algebra Worked-Out Solution Ke Mi 6. quarts of 00% antifreeze and 9.6 quarts of 0% antifreeze and 0% water mi to get 6 quarts of 70% antifreeze and 0% water mi. Eercises for the lesson Solve Linear Sstems b Substitution Skill Practice. Answers will var. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 9 The solution is about (6., 6.79) 9 () () The solution is about (0.,.). 6 6

9 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. Solve equation for, because doesn t have a coefficient in this equation. Then substitute that value into equation, and solve equation for () The solution is (, ).. () The solution is (, ).. ( ) 6 The solution is (, ) ( 7) The solution is (, 6) (6 ) () 8 0 The solution is (, ) (0 8) 0 7 0() 8 The solution is (, ) The solution is (6, 7) (8) 7 7 The solution is (8, 7) The solution is (, ) ( 9) () 9 The solution is (, ).. 9 ( ) The solution is (, 8). Algebra Worked-Out Solution Ke 9

10 (9 ) 6 6 9() The solution is (, ).. 7 ( ) () 0 The solution is (0, ) ( ) () The solution is (, ) ( 9) 0. (.) 9. The solution is (.,.). 8. A; (6, 7) ( 7) 9 6 (6) The error is that, in step, 6 should have been substituted for in the equation, not for. 9 9 (6) ( ) The solution is (, ) (0 ) () 0 0 The solution is (, 0) (7 0.) (0.).7 0. The solution is (0.,.7). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 9 Algebra Worked-Out Solution Ke

11 7. } } 8 } ( ) } } 8 } 6 } () 6 7 The solution is (, 7) (0. 9) (0) 9 } (6) 0 The solution is (0, 6). } 8. } 8 } } The solution is (, 0) } Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved..8(.8 0.) } } } } (6) The solution is (6, ). 0 } The solution is (, 0). } 9. Once ou have obtained a solution using substitution, } } } 6 } () } 6. } }8 } } } 6 } graph both equations on the same coordinate plane. The point where the lines intersect should be the same as the solution ou got using substitution. } } } } } 9 } (9) } The solution is }, 9. Algebra Worked-Out Solution Ke 9

12 The length from A to the string is represented b, which is inches. The length from the string to B is represented b, which is inches. The string should be placed inches from point A. a b 6 9a b 6 a b 6. a. d rt 9a b 6 b 6 9a 9 b } a 9a } a 6 9 d.9t k Lane swimmer d.8(t.).9t.8(t.).9t.8t.6 0.t.6 t.6 9a 6 9a 6 8a 7 a The swimmer in lane will catch up after.6 seconds. b. The race will end when the swimmers have gone 00 b } () meters. If d is less than 00 when t.6 seconds, then the swimmer in lane will catch up before the race ends. b d.9t The solution is a, b. d.9(.6) 9 d.0 meters Problem Solving. Let bags of popcorn sold. Let pretzels sold. The swimmer in lane will catch up to the swimmer in lane before the race ends.. Let ml of % hdrochloric acid solution. Let ml of % hdrochloric acid solution () 6 00 m (00) 6 0.0(00 ) (8) The sold 96 bags of popcorn and 8 pretzels. You need to mi 0 ml of the % solution and 0 ml of the % solution.. Let number of person tubes. Let number of cooler tubes. 6 m Let number of dimes. Let number of quarters. (6 ) ( ) The rented tubes for people and cooler tubes.. 9 m 9 She has quarters Let time cheetah runs..(9 ). Let time gazelle runs Algebra Worked-Out Solution Ke Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 0.

13 The cheetah would catch up to the gazelle in. seconds, but since the cheetah can onl sustain this speed for 0 seconds, the gazelle can sta ahead of the cheetah. 8. Let bushels of 00% vermiculite.. } 6 Let bushels of 60% vermiculite. 0.() ( ). 0.6 (, 7) In order to make a mi that is 60% vermiculite and 0% peat moss, he would have to add. bushels of the 00% vermiculite which would give him 6. bushels total of the miture. Since he onl needs 6 bushels, he does have enough of the 0%/0% mi. Quiz for the lessons Solve Linear Sstems b Graphing and Solve Linear Sstems b Substitution. Test (, 7). (7) 0 (, 7) is a solution () (7) The solution is (, 6).. (, ) ( ) Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved Test (, ). 0 () (.).7 The solution is (.,.7). (, ) is a solution } } ( 9) (, ) The solution is (0, 9) Test (, ). () (, ) is a solution. () () ( ) 9 () The solution is (, ). Algebra Worked-Out Solution Ke 97

14 Lesson 6. Solve Linear Sstems b Adding or Subtracting 8. 7 ( ) () Investigating Algebra Activit for the lesson Solve Linear Sstems b Adding or Subtracting. 8 The solution is (, ) (0 ) () 6 The solution is (6, ). Problem Solving Workshop for the lesson Solve Linear Sstems b Substitution Compan Compan Let number of adult tickets. Let number of student tickets. So,. 0 m 0 The solution to the sstem is (, ).. (0 ) 60 So,. Each tai will cost the same after miles adult tickets were purchased, and 70 student tickets were purchased. 98 Student Tickets Adult Tickets Total Cost ($) Algebra Worked-Out Solution Ke Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. Miles

15 So,. So,. The solution to the sstem is (, )... So,. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. So,. So, 6. The solution to the sstem is (, 6). So,. The solution to the sstem is (, ).. You cannot solve this problem using onl the steps above because neither of the variables will cancel out after ou add the equations. So ou would still have variables and be unable to solve for one of them in step. Guided Practice for the lesson Solve Linear Sstems b Adding or Subtracting. 7 () 9 The solution is (, ). Check: () () 0 () () () 0 The solution is (, ). 7 7 Algebra Worked-Out Solution Ke 99

16 7. Upstream: Check: () 6() 0 8 () () r k upstream k downstream 6() The solution is (, ).. Check: () () 0 The speed of the kaak in still water is. miles per hour, and the speed of the current is. miles per hour. 7 Eercises for the lesson Solve Linear Sstems b Adding or Subtracting 7 Skill Practice. Answers will var. 7 (). Sample answer: Subtract the equations to eliminate 7 7 the variable. Solve for. The solution is (, ). Check: 0 7() () () () (0) 6 Substitute for in Equation. Solve for. 7. The solution is (., ).. 6 The solution is (, 0) (0) 0 () () 9 The solution is (, 6). The solution is (, 9) () () The solution is (, ). The solution is (, ). Check: () () 0 Algebra Worked-Out Solution Ke () 0 () (6) () (0) Check: 00 r Let speed of current. d rt 0 r () Let speed of kaak in still water 6() () 0 d rt 0 r () The solution is (, 6). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. Downstream:

17 (7) () The solution is (, 7). 9. The solution is (, ) The solution is (, ). 6 () The solution is (6, ). 0 8 The solution is (8, ) The solution is (, ). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. The solution is (, ) The solution is (, 6). (6) 8 () The solution is (, ) (0) 0 () 8 0 The solution is (0, 0) () 6 0 The solution is (, ). The solution is (, ).. C; (, ) In this eercise the sstem must be solved b subtraction, not addition. () is 6, not. 6 8 is 8, not The solution is (, ). The solution is (, ). The solution is (, ). 7 8 (). B; (, ) 9 6 () } Algebra Worked-Out Solution Ke 0

18 . In rearranging the second equation, ou must add to both sides, making the second equation } 9 } 7 6 The solution is (6, ). 6. } } } 8 } } } } 7 } } } () } 7 8() } 8 } 6 } 6 9 The solution is (, 9). The solution is (, ) ().. () The solution is (, 8). The solution is (, ) () The solution is (, 0.).. } } } } } } } } } } () } } 0 The solution is (, ) (6) 6.8() The solution is (0, 6). The solution is (, 8).. (, ), (, ) m b a. m() b m() b b. m b m b b b m 0 m The slope is. The -intercept is. c.. w w w w (.) 9. 6w w. The length is. feet. The width is. feet. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 0 Algebra Worked-Out Solution Ke

19 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 6. Add equation and equation Then solve for to find. Substitute this value for into equation and solve for The solution to the sstem is (, ). 7. a a 6 0 a () a 0 0 The solution is (0, ). 8. Rearrange Equation so its terms are in the same order as Equation. Subtract the two equations. 7 z 9 z 7 Solve for. 9 6 Substitute for in equation. Solve for. () 0 0 Substitute these values for and into Equation and solve for z. 7() z 9 z 9 z 6 z So,,, and z. Problem Solving 9. Let speed in still water. Let speed of current The speed of the shell in still water is.6 meters per second. The speed of the current is 0. meters per second (.)..9 The fee is $.9, and the cost per quart of oil is $.0.. Let cost of a monophonic ring tone. Let cost of a polphonic ring tone A monophonic ring tone costs $.9, and a polphonic ring tone costs $.0.. a. Let number of twigs. b. Let number of flowers She used 6 twigs and flowers. Number of flowers Number of twigs Total cost ($) Algebra Worked-Out Solution Ke 0

20 . a. d rt To Phoeni: 800 r (.) 00 r The speed on the wa to Phoeni is 00 miles per hour. To Charlotte: 800 r () 0 r The speed on the wa back to Charlotte is 0 miles per hour. b. Into the wind: Speed of plane with no wind s w 00 Not into the wind: Speed of plane with no wind s w 0 s w 00 s w 0 s 80 s w 0 w Wind speed Wind speed Speed of plane Speed of plane The speed of the plane with no wind is miles per hour. The wind speed is miles per hour.. Let cost of cap-and-gown set. Let cost of an etra tassel () A cap-and-gown set costs $, and an etra tassel costs $.0. So 8 cap-and-gown sets will cost: 8() $0 And 6 etra tassels will cost: 6(.0) $96 The total the third school will spend is $0 $96 $8.. Ideal sleeve length 6.8 Ideal sleeve length Allowable deviation Allowable deviation Maimum sleeve length Minimum sleeve length The ideal sleeve length is 6. centimeters. The allowable deviation is. centimeters. Lesson 6. Solve Linear Sstems b Multipling First Guided Practice for the lesson Solve Linear Sstems b Multipling First () 6 } The solution is },. Check: 7 6 } () 0 } () (9) The solution is (9, ). Check: 9 (9) () 0 (9) 0() 0 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 0 Algebra Worked-Out Solution Ke

21 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved () 0 0 The solution is (0, ).. Let cost of soccer ball Let cost of bag (.) A soccer ball costs $.0, and the cost of a soccer ball bag is $. Eercises for the lesson Solve Linear Sstems b Multipling First Skill Practice. The least common multiple of and 8 is 6.. Multipl all of the terms in equation b so that one of the variables can be eliminated Add the two equations together to get 7. Solve for. Substitute that value into equation and solve for The solution is (, ).. () 6 The solution is (, 6) () The solution is (, ) (9) The solution is (9, 6) () 8 The solution is (, ) (8) 6 The solution is (8, ) (7) 8 6 The solution is (7, ) () 9 7 The solution is (7, ). Algebra Worked-Out Solution Ke 0

22 () The solution is (, 6) (6) 0 0 The solution is (6, 0) () 9 6 The solution is (, ) () The solution is (, ) () The solution is (, ) (7) The solution is (7, 6) () 60.9 The solution is (.9, ). 8. D; (, ) () The solution is (, ) In order to eliminate one variable, these equations must be subtracted, not added., and 8 (9) 9, so, The error is that the constants on the right sides of both equations must also be multiplied b the least common multiple () The solution is (, ). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 06 Algebra Worked-Out Solution Ke

23 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. 8 ( ) (.6) 0.9 The solution is (.6, 0.9) ( 9) (.) 9.08 The solution is (.,.08) ( ) The solution is (, ) ( ) The solution is (, ) (6.. ) (7) The solution is (, 7) (. ) () The solution is (0, ) (0). 0 0 The solution is (0, 0) () The solution is (, ) } } 0 The solution is (, )..8.8 Algebra Worked-Out Solution Ke 07

24 . } m } } m 8a b 8 a b a b 0 a b 0 6a 8 } } } } So, a, b. The solution is }, }. }. } } } 8 6. a b b a 0 6 m } } 6 m } 6 a b b a 0 m a b 8 a b } 0 b 8 b } a b 0 a b 0 a 8 } } () a6 } 6 (6) b b 8 0 b8 The solution is (0, ).. a. * w 8 m * w * w * w * 0 * () w 8 w 8 So, a 6, b 8. Problem Solving 7. Let number of hardcovers. Let number of paperbacks. m w The original length is inches, and the original width is inches. b. () 8 () 8 She purchased hardcover books. The new length is inches, and the new width is 8 inches.. a 6 If a ou can solve the sstem b substitution. You can solve the first equation for and substitute for in the second equation. If a 0 ou can solve the sstem b solving the first equation for and substitute that value into the second equation. If a or a, ou can solve the sstem b addition or subtraction since the second equation contains the term. That term could be eliminated without use of multiplication.. a b 0, 8. Let cost for song. Let cost for album..9 m (0.99) The website charges $.99 to download a song and $9.99 to download an album. 9. Let number of pies. b a 0 Let batches of applesauce. a b 69 b a m Algebra Worked-Out Solution Ke Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. a () b

25 () m You can make pies and 6 batches of applesauce a. Let student tickets sold Let adult tickets sold. 800 m He invested $800 in stocks and $00 in bonds.. Let t time to reach destination. d 808 d rt, t }r 0 t going t returning t total 0 79 b. 7 r 60 r 7 0 r 7 }} } r }r There were tickets sold to students and 0 tickets sold to adults. }}} }} } r 60 r } (60) The average speed going is 60 miles per hour. The average speed returning is miles per hour. 79 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 00 The solution in part (a) is reasonable. Quiz for the lessons Solve Linear Sstems b Adding or Subtracting and Solve Linear Sstems b Multipling First.. Let cost of small dish. 8 Let cost of large dish. 8.0 m m The solution is (, ). (.9) The cost of a small dish is $.90 and the cost of a large dish is $.90. The cost of small dishes can be found b: (.90) $8.70 () The cost of large dishes can be found b: (.90) $7.80 The total cost is, The solution is (, ). $8.70 $7.80 $6.0.. Answers will var... Let amount invested in stocks. Let amount invested in bonds. The solution is (, ) () 0 The solution is (, ). Algebra Worked-Out Solution Ke 09

26 . 0 0 m () The solution is (.9,.7) (.6) The solution is (.,.6). m (0.7) m 9 } m 9 } 6 0 } () } 6 The solution is (6, ). Graphing Calculator Activit for the lesson Solve Linear Sstems b Multipling First Practice.08 The solution is (0.7,.08). 8. m Equation Equation Step Solve both equations for. 9 9 Graph the two equations using a graphing calculator. Notice that the point of intersection of the graphs is the solution of the sstem. The solution is (, ) The solution is (0, 9) m 6 0 m () Add the two equations as ou would if ou were solving the sstem algebraicall. Graph the resulting equation. 9 The solution is (, ).. 7 m 6 9 m Algebra Worked-Out Solution Ke Step The solution is (, ). Equation Equation Add. Solve for. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. } The solution is (, ).

27 Now graph the equation on the same calculator screen with the two original equations. Now graph the equation on the same calculator screen with the two original equations. Step Step All three equations intersect at (, ). All three equations intersect at (, ). So, (, ) is the solution of the sstem. So, (, ) is the solution of the sstem Equation Equation Step Equation Step Graph the two equations using a graphing calculator. Notice that the point of intersection of the graphs is the solution of the sstem } } 7 } 7 } 7 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Equation 8 Solve both equations for. Solve both equations for } Graph the two equations using a graphing calculator. Notice that the point of intersection of the graphs is the solution of the sstem. The solution is (, ). The solution is },. Step Step Add the two equations as ou would if ou were solving the sstem algebraicall. Graph the resulting equation. Add the two equations as ou would if ou were solving the sstem algebraicall. Graph the resulting equation. 6 7 Equation Equation Equation 8 Equation Add. 7 7 Equation 7 7 Substitute for. } Solve for. Net solve for when in one of the equations. 7 0 Subtract from each side. Divide each side b 7. Add. Solve for. Net solve for when } in one of the equations. Equation Substitute } for. } Simplif. Subtract from each side. Divide each side b. Algebra Worked-Out Solution Ke

28 Now graph the equation on the same calculator screen with the two original equations. Step All of the equations intersect at (, ). So, (, ) is the solution of the sstem.. Equation Equation Step Solve each equation for. Step All three equations intersect at },. So, }, is the solution of the sstem. } } Graph the two equations. The point of intersection of the graphs is the solution of the sstem. Practice. Equation Equation Step Solve each equation for. } Graph the two equations. The point of intersection of the graphs is the solution of the sstem. The graphs intersect at (, ). Step Multipl Equation b so that ou can eliminate the variable b adding. n + Step Graph the equations,, and on the same graphing calculator screen with the original equations. The graphs intersect at (, ). Step Multipl Equation b so that ou can eliminate the variable b adding. + 0 n 7 Step Graph the equations 0,, and on the same graphing calculator screen with the original equations. Step All of the equations intersect at (, ). So, (, ) is the solution of the sstem. 6. Equation 9 Equation Step Solve each equation for. Algebra Worked-Out Solution Ke 9 9 } 9 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.

29 Graph the two equations. The point of intersection of the graphs is the solution of the sstem. The solution using the graphing calculator is (, ). Multipl Equation b so that ou can eliminate the variable b adding. 6 n Multipl Equation b so that ou can eliminate the variable b adding. The graphs intersect at (6, ). 6 n Step 8 n Step Graph the equations, 7, and on the same graphing calculator screen with the original equations. 6 0 Multipl Equation b so that ou can eliminate the variable b adding. 6 The solution (, ) found using the graphing calculator has coordinates that are given b the equations resulting from using linear combinations to eliminate one of the variables from the original sstem. Draw Conclusions 8. To check the solution of the sstem, ou can add the equations and graph the sum. If the three lines intersect in one point, the solution is correct. 9. a. You get a false equation such as. b. The lines are parallel. c. When ou add the equations, ou don t get an Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. equation that can be graphed, so the method does not work. Step All of the equations intersect at (6, ). So, (6, ) is the solution of the sstem Equation 8 Equation Solve each equation for } Graph the two equations. The point of intersection of the graphs is the solution of the sstem. Mied Review of Problem Solving for the lessons Solve Linear Sstems b Graphing, Solve Linear Sstems b Substitution, Solve Linear Sstems b Adding or Subtracting, and Solve Linear Sstems b Multipling First. Let speed in still air. Let speed in wind. a. } 60, } The average speed for the first flight is 60 km/h, and 7 km/h for the return. b c The helicopter s average speed in still air is 67. kilometers per hour. The speed of the wind is 7. kilometers per hour.. Let cost of a pound of potato salad. Let cost of a pound of coleslaw Algebra Worked-Out Solution Ke

30 (6..) (.7). (.) 6.0 (.7) Two pounds of potato salad cost $6.0, and two pounds of coleslaw cost $.0, a total of $.. The point represented b the intersection of the graphs shows that the customers will have paid the same amount after months.. Answers will var.. d rt h d 700(0.) 70 0 d 000(0.) The balloon at Kirb Park will ascend to a height of 0 feet. The balloon at Newman Park will ascend to a height of 90 feet with regards to sea level. After about minutes, the two balloons will be at the same distance from sea level. 6. a (00) b (00 ) To make the 0% acid and 80% water mi, 0 milliliters of the 0% acid and 90% water mi is combined with 0 milliliters of the 0% acid and 70% water mi. c (00) 0.(00 ) The chemist does need more of the 0% acid and 90% water mi because this new mi has less acid than the first mi created, so more of the less acidic solution must be added. Lesson 6. Solve Special Tpes of Linear Sstems Guided Practice for the lesson Solve Special Tpes of Linear Sstems This is a false statement. Since the variables are eliminated and ou are left with a false statement, regardless of the values of and. This tells ou that the sstem has no solution ( ) 6 6 The variables are eliminated and ou are left with a statement that is true regardless of the values of and. This tells ou that the sstem has infinitel man solutions } } 6 The sstem has one solution.. Let cost of regular print. 0( ) A gloss print costs $8.00. Eercises for the lesson Solve Special Tpes of Linear Sstems Skill Practice. A linear sstem with no solution is called an inconsistent sstem.. A linear sstem with infinitel man solutions is called a consistent dependent sstem.. The graph of a linear sstem with no solution is two parallel lines.. The graph of a linear sstem with infinitel man solutions is two lines that coincide producing a graph that appears to be a single line } Matches graph B. Sstem has one solution. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

31 6. } 6 0 Matches graph C. Sstem has no solution } } } } Matches graph A. Sstem has infinitel man solutions. 8. The sstem has one solution } } The sstem has no solution. The sstem has infinitel man solutions. } 9.. } } } 6 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. The sstem has no solution The sstem has infinitel man solutions }. The error is that, though the lines do not intersect in the graph shown, the are not parallel, so the will intersect at some point. Therefore the sstem has one solution.. m () 6 9 The solution is (, ) The sstem has one solution } 8 6 (8 ) 6 6 The sstem has infinitel man solutions. Algebra Worked-Out Solution Ke

32 7. 7 m 8 0 m 6. C; 9. D; (7) 6 9 The sstem has no solution The solution is (, 7) The sstem has infinitel man solutions. ( 0.6) m } 9 The sstem has one solution } } 0 8 } } The sstem has infinitel man solutions The solution is (, ) } m 8 6 }6 } The sstem has one solution The sstem has no solution }7 } 7 (8 ) }7 } 7 0 8(0) The sstem has infinitel man solutions.. Answers will var. The solution is (, 0).. Sample answer: m () 9 p q r a. p, q, r Þ 0 b. p, q, r Algebra Worked-Out Solution Ke 0. Sample answer: 6 0 The solution is (0, ).. Answers will var. c. p, q, r Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved The sstem has no solution. 6 7 } 9 The sstem has infinitel man solutions. 0. 7

33 Problem Solving 0. a m There are infinitel man solutions, so ou cannot determine the cost of one admission to the roller skating rink. You need more information. 7. Let cost of a coach ticket. Let cost of a business class ticket. 0 80, , (7.8.7),860 0,8 6, (7) 7 There is enough information to determine that one coach ticket costs $7. 8. a. Let memor needed for a picture. Let memor needed for a 0 second movie Etension for the lesson Solve Special Tpes of Linear Sstems.. b. Let memor used for a picture.. Now ou can determine that a 0 second movie uses megabtes of memor. 9. a. d rt 0 d } t m d } t 0 d } (t ) m d } t d The lines are parallel, so the will never instersect. The second climber will never catch up to the first. d t t {, if <, if > 0, if 0 <, if <, if < 6., if < { 7. P { 0t, if 0 t 0 0(t 0) 800, if t > 0 P 00 Pa (dollars) 0(0.08) d t {,, ifif > 00 laate-bkm-ala-a06, if laate-bkm-ala-a0 0 7(0) 6 b. laate-bkm-ala-a0 0 memor used for a movie There are infinitel man solutions, so ou cannot determine the amount of memor used b a 0 second movie. You need more information.. m Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. ( 0) 0 b. 0 0 (0) 00 c. The solution does not make sense because ou cannot fold a negative number of napkins.. Let d distance from Salem to Lancaster. Let wind speed. d rt d (60 ) d 0 (60 ) (60 ) 0 (60 ) d (60 0) d 0 It is 0 miles from Salem to Lancaster t Time (hours) When t 6, P $980. laate-bkm-ala-a07 Algebra Worked-Out Solution Ke 7

34 8. The parent absolute value funtion is defined b two 9. equations. For negative values of, and, if < 0 for nonnegative values of, ;., if 0 {. b. < b > } 6. b 8 m r 8 r 6 b 0 r0 0 Since the point does fall in the shaded area, this bat can be used b a senior league plaer. Yes; es; the function is a piecewise function because, for ever integer value of n, there is a unique equation that applies to the part of the domain defined b laate-bkm-ala-a08 n < n. The function is a step function because it is defined b a constant value over each part of its domain. Lesson 6.6 Solve Sstems of Linear Inequalities Guided Practice for the lesson Solve Sstems of Linear Inequalities. < r Eercises for the lesson Solve Sstems of Linear Inequalities Skill Practice. A solution of a sstem of linear inequalities is an ordered pair that is a solution of each inequalit in the sstem.. Graph both inequalities in the same coordinate plane. Use a dashed line for Inequalit and a solid line for Inequalit. Find the intersection of the half-planes and shade that area.. (, ) is not a solution.. (0, 6) is a solution.. (, ) is not a solution. 6. Matches graph C. 7. Matches graph A. 8. Matches graph B r <. <.. r. < >. 8 Algebra Worked-Out Solution Ke Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.

35 . b 8, < 8 6. > 6, r b 8 r }. D; (, ) > 6 b, > > } Tr (, ). b ò Tr (, ). 7 b ò Tr (, ). 6> Tr (, 0). b 7>. B; < 7. < 7 > < 7 > > 6 Check (, ). < () () () > 6 <6 9>6. Inequalit sas that is less than, not greater than, so the shaded region should be below that line, rather than above.. >, <. >, < 6. r, < 8. <, >, r 7. > b 8. b 0 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 9. > 6 r b b > r0 0. >, < 9, 9. r 0, r 0, 6 < b > < 6 > 6. < > < 0. b 0 r b 0 r > b r r b b r Algebra Worked-Out Solution Ke 9

36 . b 0 r 9 8. a. Let surfperch. 9 r } } b b b 6 Let rockfish. r 0, r 0 b, b b r } r b s b. (, 9) You cannot catch surfperch and 9 rockfish in one da. 9. a. r b 6. No, the sstem has no solutions because no point makes r 0.7(0 ) b 0.8(0 ) r 0.7 b both inequalities true. 00. r r b6 b r } 0 r } 00 b } 0 Problem Solving 6. r 0 music score b music score r0 visual score b visual score b. B finding the points on the graph, ou can see that his heart rate does not sta in the suggested range. It falls below it. 0. Let in. b in. pictures. Let in. b 6 in. pictures. a. r 6 m r b } 7 < 70 6 (, 6) 8 7 < 70 b. Because the point falls within the solution region, < 0 ou are able to bu pictures that are inches b inches and 6 pictures that are inches b 6 inches. Algebra Worked-Out Solution Ke b 8 7. r r < 8, 0 8 } } b 8 0 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved..

37 . a } } } 6 6 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 0 0 b. (0, 0), (0, 0), (0, 0), (0, 0) c. R 0 8 R 0(0) 8(0) 0 R 0(0) 8(0) 0 R 0(0) 8(0) 60 R 0(0) 8(0) 00 The verte (0, 0) results in the maimum revenue. Quiz for the lesson Solve Special Tpes of Linear Sstems and Solve Sstems of Linear Inequalities The sstem has no solution The sstem has infinitel man solutions.. > < 7. < 6 6. < < 7. < > < } < > 0 The sstem has one solution. Algebra Worked-Out Solution Ke

38 8. r b c. r ( ) A gallon of wall paint is $9. A gallon of ceiling paint is $.. Let cost of a CD. Let cost of a DVD >, < 0, b () m There are infinitel man solutions, so ou need more information to determine the cost of one CD.. The graph shows that the two lines are parallel, which means the will never intersect. So, bus B will never catch up to bus A.. a. Let hours lifeguarding. Let hours working at a retail store. 0 8 r 00 b 0 r } b 0. a. Let number of bricks. Let number of bags of sand. r 600, r 0 b 000 b. 0 b 000 b } 60 b. B locating the point (, ) on the graph, ou can see it does not lie in the shaded region. So; ou will not earn at least $00 per week. c. You can work to 0 hours as a lifeguard to earn at least $00 per week Answers will var. 7. r 0, 00 r0 b c. 700 bricks and 0 bags of sand can be delivered in (0) 60 one trip. k -intercept. Let cost of a gallon of ceiling paint. (0) 60 Let cost of a gallon of wall paint. a k -intercept 0 () m 0 00 b. There is not enough information to determine the cost of each tpe of paint. There are infinitel man solutions, so ou need more information. Algebra Worked-Out Solution Ke Area } bh, b, h A } ()() 90 The area of the triangle is 90 square feet. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Mied Review of Problem Solving for the lessons Solve Special Tpes of Linear Sstems and Solve Sstems of Linear Inequalities

39 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Chapter Review for the chapter "Sstems of Equations and Inequalities". A sstem of linear inequalities consists of two or more linear inequalities in the same variables.. A sstem of linear equations consists of two or more linear equations in the same variables.. Graph each of the inequalities on the same coordinate plane. Shade the intersection of the two half-planes. To check our solution, choose a point in the shaded region and substitute its and -values into each inequalit.. Answers will var.. 7 (, ) Check (, ). 7 0 () 0 7 The solution is (, ). 6. (, ) Check (, ). 0 () 0 () The solution is (, ). 7. Check (, ). (, ) () 0 () 0 The solution is (, ) ( 7) 9 () 7 The solution is (, ). The solution is (, ) () 6 9 The solution is (, 9).. Let tubes of paint. Let brushes () () 8 She purchases tubes of paint and 8 brushes () 0 The solution is (, ). The solution is (, ) (6) The solution is (, ). The solution is (6, 0). Algebra Worked-Out Solution Ke

40 (8) 9 (8) The solution is (8, ). 7 The solution is (7, 8) The solution is (7, ) () 8 The solution is (, ) () The solution is (9, ) () 6 The solution is (, ) () The solution is (, ) () 6 The solution is (, 6).. Let cost of gallon of gasoline. Let cost of quart of oil (.).0 One quart of oil costs $ } } } The lines are parallel because the have the same slope but different -intercepts. Parallel lines do not intersect, so the sstem has no solution The lines are the same, so the sstem has infinitel man solutions The lines have different slopes and different -intercepts, so the intersect at one point. Therefore the sstem has one solution. 8. < > Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

41 9. b. > (, ) 0. r 0, b < Check (, ). () 0 0 The solution is (, ).. ( ) } 6 Let number of evening tickets. r0,. Let number of matinee tickets. r 0, 8 b 0 8 b 0 b }8 Check (0.6,.).. 0 (0.6) (0.6) (.) 0. <..98 < Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. The solution is (0.6,.). Chapter Test for the chapter "Sstems of Equations and Inequalities" } } (, ) 6 (, 0) 0 Check (, ). () The solution is (, ). 0 Check (, 0). () (0) (0) 0 The solution is (, 0). Algebra Worked-Out Solution Ke

42 } (, ) Check (, ). 9 () 0 9 () The solution is (, ) (7, ) } 7 } Check (7, ). (7) 7() 0 (7) 7() The solution is (7, ) (6) The solution is (6, ) ( 7) The solution is (, 0). Algebra Worked-Out Solution Ke The solution is (7, 6) ( 70) () 70 0 The solution is (, 0) ( 8) The solution is (.,.) (0. 9) (7.7) 9.7 The solution is (7.7,.7) () The solution is (, ) The solution is (7, ) The solution is (, ). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.

43 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved The solution is (, 7) (7) 6 The solution is (6, 7) () 9 6 The solution is (, ). 9. ( ) 0 0 The sstem has infinitel man solutions The sstem has no solution } The sstem has one solution The sstem has no solution (8) 0 The sstem has one solution The sstem has infinitel man solutions.. < > > (.) 0 The fee is $0, and the cost per mile is $.0. Algebra Worked-Out Solution Ke 7

44 9. w 8 w (w ) w 8 6w w 8 8w 6 w 7 (7) The length is inches and the width is 7 inches. 0. Let number of granola bars purchased. Let number of bottles of water purchased. 0, Etra Practice for the chapter Sstems of Equations and Inequalities. The graphs appear to intersect at (, ). 0 0 Because (, ) is a solution of each equation, it is a solution of the linear sstem.. The graphs appear to intersect at (, 6). 6 0 () 6 0 () Because (, 6) is a solution to each equation, it is a solution of the linear sstem.. The graphs appear to intersect at (, ). () 0 () 0 Because (, ) is a solution of each equation, it is a solution of the linear sstem The graphs appear to intersect at (, 6). 0 () Because (, 6) is a solution of each equation, it is a solution of the linear sstem.. } } 8 } 6 8 The graphs appear to intersect at (, ). 8 () () 0 () () Because (, ) is a solution of each equation, it is a solution of the linear sstem. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8 Algebra Worked-Out Solution Ke

45 6. } } } 6 } } } } ( 0) The solution is (, 7). Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 0 6 The graphs appear to intersect at (6, ). } } } 6 } } 8 } (6) } () 0 } 6 } (6) 0 } 8 6 } } 6 8 } } 8 Because (6, ) is a solution of each equation, it is a solution of the linear sstem ( 6) () 6 0 The solution is (, 0). 8. ( ) } } } The solution is }, }. 9. ( ) 7 The solution is (9, 7). (7) () 6 8 (8) 6 The solution is (8, 6).. } } } 6 } 6 } } 6 } The solution is (7, ).. () The solution is (, ). } () The solution is (, ) (9) The solution is (9, ) () 6 7 } The solution is, } 7. 8 Algebra Worked-Out Solution Ke 9

46 m 0 m 6. m m () 6 (0) } The solution is (, ). m 8. 9 The solutin is 0, } m m m (0) 9 } 9 } 8 6 } The solution is }, m 9 } 7 9 m,}. The solution is } m 6 The solution is (6, ). 9 m m The lines are parallel so this linear sstem has no solutions. () 9 6. The solution is (, ). m m m 8 m (9) 6 0 The solution is (, 9).. m m () } The lines appear to intersect at the point (, ), so the linear sstem has one solution m 0 The solution is },. 0 Algebra Worked-Out Solution Ke Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 0.

47 The equations represent the same line so the linear sstem has infinitel man solutions ( ) 0 The solution is (0, ) ( ) 6 0 The solution is (0, 8). (0) Onl a false statement is left. This tells ou the sstem has no solution } 7 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 6 } 7 Onl a false statement remains. This tells ou the sstem has no solution.. ( ) Onl a false statement remains. This tells ou the sstem has no solution.. 0. } } 0 0 The solution is (0, 0). } (0) 0. Algebra Worked-Out Solution Ke