Systems in Three Variables. No solution No point lies in all three planes. One solution The planes intersect at one point.

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1 3-5 Systms in Thr Variabls TEKS FOCUS VOCABULARY TEKS (3)(B) Solv systms of thr linar quations in thr variabls by using Gaussian limination, tchnology with matrics, and substitution. Rprsntation a way to display or TEKS (1)(D) Communicat mathmatical idas, rasoning, and thir implications using multipl rprsntations, including symbols, diagrams, graphs, and languag as appropriat. dscrib information. You can us a rprsntation to prsnt mathmatical idas and data. Additional TEKS (1)(A), (3)(A) ESSENTIAL UNDERSTANDING To solv systms of thr quations in thr variabls, you can us som of th sam algbraic mthods you usd to solv systms of two quations in two variabls. Ky Concpt Solutions of Systms With Thr Variabls No solution No point lis in all thr plans. On solution Th plans intrsct at on point. An infinit numbr of solutions Th plans intrsct at all th points along a common lin. Problm 1 Solving a Systm Using Elimination Which variabl do you liminat first? Eliminat th variabl for which th procss rquirs th fwst stps. What is th solution of th systm? Us limination. Th quations ar numbrd to mak th procdur asy to follow. ① 2x y z 4 ② x 3y z 11 ③ 4x y z 14 Stp 1 Pair th quations to liminat z. Thn you will hav two quations in x and y. Add. Subtract. ② ① 2x - y + z = 4 x + 3y - z = 11 ③ 4x + y - z = 14 ② x + 3y - z = 11 ④ 3x + 2y ⑤ -3x + 2y = 15 = -3 continud on nxt pag 96 Lsson 3-5 Systms in Thr Variabls

2 Problm 1 continud Dos it mattr which quation you substitut into to find z? No, you can substitut into any of th original thr quations. Stp 2 Writ th two nw quations as a systm. Solv for x and y. Add and solv for y. Substitut y = 3 and solv for x. 4 3x + 2y = x + 2y = -3 Stp 3 4y = 12 y = 3 4 3x + 2y = 15 3x + 2(3) = 15 3x = 9 x = 3 Solv for z. Substitut th valus of x and y into on of th original quations. 1 2x - y + z = 4 2(3) z = z = 4 z = 1 Us quation 1. Substitut. Simplify. Solv for z. Stp 4 Writ th solution as an ordrd tripl. Th solution is (3, 3, 1). Problm 2 TEKS Procss Standard (1)(D) Solving an Equivalnt Systm What is th solution of th systm? Us limination. 1 x y 2z 3 2 2x y 3z 7 3 x 2y z 10 You ar trying to gt two quations in x and z. Multiply 1 so you can add it to 2 and liminat y. Do th sam with 2 and 3. Multiply 4 so you can add it to 5 and liminat x. Substitut z = 3 into 4. Solv for x. Substitut th valus for x and z into 1 to find y. Chck th answr in th thr original quations. 1 2 x y 2z 3 2x y 3z 7 2 2x y 3z 7 3 x 2y z b x z 4 3x 7z 24 x 3 4 x 1 x y 2z 3 1 y 2(3) 3 y 4 Th solution is (1, 4, 3). 4 5 x y 2z 3 2x y 3z 7 x z 4 4x 2y 6z 14 x 2y z 10 3x 7z 24 3x 3z 12 3x 7z 24 4z 12 z 3 Chck 1 ( 4) 2(3) 3 2(1) ( 4) 3(3) 7 (1) 2( 4) 3 10 ParsonTEXAS.com 97

3 Problm 3 Solving a Systm Using Substitution Multipl Choic What is th x-valu in th solution of th systm? 1 2x 3y 2z 1 2 x 5y 9 3 4z 5x 4 Which quation should you solv for on of its variabls? Look for an quation that has a variabl with cofficint Stp 1 Choos quation 2. Solv for x. Stp 2 2 x + 5y = 9 x = 9-5y Substitut th xprssion for x into quations 1 and 3 and simplify. 1 2x + 3y - 2z = z - 5x = 4 2(9-5y) + 3y - 2z = -1 4z - 5(9-5y) = y + 3y - 2z = -1 4z y = y - 2z = -1 4z + 25y = y - 2z = y + 4z = 49 Stp 3 Writ th two nw quations as a systm. Solv for y and z. 4-7y - 2z = y + 4z = 49-14y - 4z = -38 Multiply by 2. 25y + 4z = 49 Thn add. 11y = 11 y = 1 4-7y - 2z = -19-7(1) - 2z = -19 Substitut th valu of y into 4. -2z = -12 z = 6 Stp 4 Us on of th original quations to solv for x. 2 x + 5y = 9 x + 5(1) = 9 Substitut th valu of y into 2. x = 4 Th solution of th systm is (4, 1, 6), and x = 4. Th corrct answr is B. 98 Lsson 3-5 Systms in Thr Variabls

4 Problm 4 TEKS Procss Standard (1)(A) Solving a Ral-World Problm Businss You manag a clothing stor and budgt $6000 to rstock 200 shirts. You can buy T-shirts for $12 ach, polo shirts for $24 ach, and rugby shirts for $36 ach. If you want to hav twic as many rugby shirts as polo shirts, how many of ach typ of shirt should you buy? Rlat T-shirts + polo shirts + rugby shirts = 200 How many unknowns ar thr? Thr ar thr unknowns: th numbr of ach typ of shirt. rugby shirts = 2 # polo shirts 12 # T-shirts + 24 # polo shirts + 36 # rugby shirts = 6000 Dfin Lt x = th numbr of T-shirts. Lt y = th numbr of polo shirts. Lt z = th numbr of rugby shirts. Writ 1 x + y + z = c z = 2 # y 3 Stp 1 Stp 2 12 # x + 24 # y + 36 # z = 6000 Sinc 12 is a common factor of all th trms in quation 3, writ a simplr quivalnt quation. 3 12x + 24y + 36z = x + 2y + 3z = 500 Divid by 12. Substitut 2y for z in quations 1 and 4. Simplify to find quations 5 and 6. 1 x + y + z = x + 2y + 3z = 500 x + y + (2y) = 200 x + 2y + 3(2y) = x + 3y = x + 8y = 500 Stp 3 Writ 5 and 6 as a systm. Solv for x and y. 5 x + 3y = 200 -x - 3y = -200 Multiply by x + 8y = 500 x + 8y = 500 Thn add. 5y = 300 y = 60 5 x + 3y = 200 x + 3(60) = 200 Substitut th valu of y into 5. x = 20 Stp 4 Substitut th valu of y in 2 and solv for z. 2 z = 2y z = 2(60) = 120 You should buy 20 T-shirts, 60 polo shirts, and 120 rugby shirts. ParsonTEXAS.com 99

5 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan pag for a Virtual Nrd tutorial vido. For additional support whn complting your homwork, go to ParsonTEXAS.com. Solv ach systm by limination. Chck your answrs x - y + z = -1 x + y + 3z = -3 2x - y + 2z = 0 a + b + c = -3 3b - c = 4 2a - b - 2c = x - y - 2z = 4 -x + 2y + z = 1 -x + y - 3z = 11 6q - r + 2s = 8 2q + 3r - s = -9 4q + 2r + 5s = x + y - z = 2 -x - 3y + z = -10 3x + 6z = -24 x - y + 2z = -7 y + z = 1 x = 2y + 3z STEM 7. x + 2y = 2 2x + 3y - z = -9 4x + 2y + 5z = x + 2y + 2z = -2 2x + y - z = -2 x - 3y + z = 0 9. x + 4y - 5z = -7 3x + 2y + 3z = 7 2x + y + 5z = Apply Mathmatics (1)(A) In a factory thr ar thr machins, A, B, and C. Whn all thr machins ar working, thy produc 287 bolts pr hour. Whn only machins A and C ar working, thy produc 197 bolts pr hour. Whn only machins A and B ar working, thy produc 202 bolts pr hour. How many bolts can ach machin produc pr hour? 11. In PQR, th masur of angl Q is thr tims that of angl P. Th masur of angl R is 20 mor than that of angl P. Find th masur of ach angl. 12. Apply Mathmatics (1)(A) A stadium has 49,000 sats. Th numbr of sats in Sction A quals th total numbr of sats in Sctions B and C. Suppos th stadium taks in $1,052,000 from ach sold-out vnt. How many sats dos ach sction hold? A $25 VISITORS C B B $15 $20 $20 C $15 HOME A $25 Solv ach systm by substitution. Chck your answrs. 13. x + 2y + 3z = 6 y + 2z = 0 z = a + b + c = 7 a + 3b - c = 13 b = 2a r - 4s - 3t = 3 t = s + r r = 3s = 3x - y 4y - 3x + 2z = -3 z = 2x - 4y 17. x + 3y - z = -4 2x - y + 2z = 13 3x - 2y - z = x - 4y + z = 6 2x + 5y - z = 7 2x - y - z = Lsson 3-5 Systms in Thr Variabls

6 Solv ach systm using any mthod. 19. x - 3y + 2z = 11 -x + 4y + 3z = 5 2x - 2y - 4z = x + 2y + z = 4 2x - y + 4z = -8-3x + y - 2z = x - y + 2z = -6-2x + 3y - z = 8 2y + 3z = x - 2y + 5z = 6 3x + 3y + 8z = 4 x - 5y - 3z = / + 2w + h = 72 / = 3w h = 2w 24. 6x + y - 4z = -8 y 4 - z 6 = 0 2x - z = Apply Mathmatics (1)(A) A workr rcivd a $10,000 bonus and dcidd to split it among thr diffrnt accounts. H placd part in a savings account paying 4.5% pr yar, twic as much in govrnmnt bonds paying 5%, and th rst in a mutual fund that rturnd 4%. His incom from ths invstmnts aftr on yar was $455. How much did th workr plac in ach account? 26. Connct Mathmatical Idas (1)(F) Writ your own systm with thr variabls. Bgin by choosing th solution. Thn writ thr quations that ar tru for your solution. Us limination to solv th systm. 27. Rfr to th rgular fiv-pointd star at th right. Writ and solv a systm of thr quations to find th masur of ach labld angl. 28. In th rgular polyhdron dscribd blow, all facs ar congrunt polygons. Us a systm of thr linar quations to find th numbrs of vrtics, dgs, and facs. Evry fac has fiv dgs and vry dg is shard by two facs. Evry fac has fiv vrtics and vry vrtx is shard by thr facs. Th sum of th numbr of vrtics and facs is two mor than th numbr of dgs. x z y y TEXAS Tst Practic y = - 2x What is th valu of z in th solution of th systm? - x + y - 2z = - 2 3x - 2y + 4z = What is th x-intrcpt of th lin at th right aftr it is translatd up 3 units? 31. Suppos y varis dirctly with x, and y = 15 whn x = 10. What is y whn x = 22? 32. A thatr has 490 sats. Sats sll for $25 on th floor, $20 in th mzzanin, and $15 in th balcony. Th numbr of sats on th floor quals th total numbr of sats in th mzzanin and balcony. Suppos th thatr taks in $10,520 from ach sold-out vnt. How many sats dos th mzzanin sction hold? y 2 O 2 2 x ParsonTEXAS.com 101

7 Activity Lab Graphs in Thr Dimnsions Us With Lsson 3-5 tks (1)(A) To dscrib positions in spac, you nd a thr-dimnsional coordinat systm. You hav larnd to graph on an xy-coordinat plan using ordrd pairs. Adding a third axis, th z-axis, to th xy-coordinat plan crats coordinat spac. In coordinat spac you graph points using ordrd tripls of th form (x, y, z). origin Points in a Plan y-axis O (2, 3) 3 units up x-axis 2 units right ordrd pair A two-dimnsional coordinat systm allows you to graph points in a plan. origin 2 units forward Points in Spac z-axis (2, 3, 4) O ordrd tripl 4 units up y-axis x-axis 3 units right A thr-dimnsional coordinat systm allows you to graph points in spac. In th coordinat plan, point (2, 3) is two units right and thr units up from th origin. In coordinat spac, point (2, 3, 4) is two units forward, thr units right, and four units up. Dfin on cornr of your classroom as th origin of a thr-dimnsional coordinat systm lik th classroom shown. Writ th coordinats of ach itm in your coordinat systm. 1. ach cornr of your classroom 2. ach cornr of your dsk 3. on cornr of th blackboard 4. th clock 1 5. th wast-papr baskt 6. Pick 3 itms in your classroom and writ th coordinats of ach. z x y continud on nxt pag 94 Activity Lab Graphs in Thr Dimnsions

8 Activity Lab continud An quation in two variabls rprsnts a lin in a plan. An quation in thr variabls rprsnts a plan in spac. 2 Givn th following quation in thr variabls, draw th plan in a coordinat spac. x + 2y z = 6 7. Lt x = 0. graph th rsulting quation in th yz-plan. 8. Lt y = 0. graph th rsulting quation in th xz-plan. From gomtry you know that two non-skw lins dtrmin a plan. 9. Sktch th plan x + 2y - z = 6. (If you nd hlp, find a third lin by ltting z = 0 and thn graph th rsulting quation in th xy-plan.) 3 Two quations in thr variabls rprsnt two plans in spac. 10. Draw th two plans dtrmind by th following quations: 2x + 3y - z = 12 2x - 4y + z = Dscrib th intrsction of th two plans abov. Exrciss Find th coordinats of ach point in th diagram. 12. A 13. B 14. C 15. D 16. E 17. F Sktch th graph of ach quation. 18. x - y - 4z = x + y + z = x + 5y + 10z = x + 6y - 12z = 36 Graph th following pairs of quations in th sam coordinat spac and dscrib thir intrsction, if any x + 3y + z = x - 3y + 5z = 7-3x + 5y - 2z = 60 2x - 3y - 4z = -4 ParsonTEXAS.com 95