MAT 080Algebra II. Literal Equation


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1 MAT 080Algeba II Lieal Equaions Objecives a Solve (linea) lieal equaions which o no equie facoing b Solve (linea) lieal equaions which equie facoing a Solving lieal equaions which o no equie facoing A lieal equaion, o a fomula, is jus an equaion conaining moe han one vaiable. Mos hings efee o as fomulas ae lieal equaions. Fo example: The fomula fo simple inees: I p The fomula fo he peimee of a ecangle: P w The fomula elaing isance, ae an ime fo moion: Now in Algeba I you leane how o solve equaions like 3x 5 8, which have one single vaiable (in his case, x). An impoan pincipal in algeba is: anyhing ha you can o wih numbes will wok wih vaiables. So if I eplace he 3, 5 an 8 wih, say,, s an, especively, o give x s, I shoul sill be able o solve his fo x. Look a he following wok, one in paallel beween he wo poblems: Equaion Lieal Equaion 3x 5 8 oiginal poblem x s 3x move he em wihou x x s s s o he ohe sie 3x 3 subac x s 3x ivie o eliminae he muliplie faco, leaving jus x x 1 cancel, leaving he answe x 1 x s s
2 MAT 080: Lieal Equaions Noe ha wheneve we subace a numbe o eliminae an ae em in he equaion, we subace a vaiable o eliminae a vaiable em in he lieal equaion. An wheneve we ivie by a numbe o eliminae a muliplie faco in he equaion, we ivie by a vaiable em o eliminae a muliplie faco in he lieal equaion. In ohe wos: o solve a lieal equaion fo a vaiable, you will use he same poceues ha you use when solving an equaion fo is vaiable. The iffeence is ha you will nee o apply hese poceues o vaiable ems (as well as o numbes) when solving a lieal equaion. Example 1: Solve fo. Soluion: Noe fis ha I neee o ell you which vaiable o solve fo because hee ae hee vaiables in he fomula. To ge by iself you nee o ge i of he, which muliplies he. To ge i of muliplie quaniies in an equaion, you nee o ivie. Given lieal equaion Divie by o cancel he an isolae he. Cancel The answe o he poblem is. You may fin his somewha o: befoe, when you solve an equaion, he answe woul be a numbe, no an expession wih lees like his answe. Bu he oiginal poblem ha hee vaiables in i, so one shoul expec ha he answe will have hee vaiables, as well. Thee ae wo hings ha nee o be ue fo o be he soluion o he oiginal lieal equaion: (he vaiable you solve fo) nees o be by iself on one sie of he equal sign, an, canno appea on he ohe sie of he equal sign. Pacice Poblem 1: Solve fo. The soluion o his Pacice Poblem may be foun saing on page 10.
3 Example Solve A x y z fo z. Objecive a: Solving lieal equaions which o no equie facoing 3 Soluion: We ae o solve fo z, which means we nee o ge he z by iself on one sie of he equal sign. To ge he z by iself, you nee o ge i of boh he x an he y, which ae ae o he z. To ge i of ae quaniies in an equaion, you nee o subac. A x y z A x y x y z x y A x y x x y y z A x y z Given lieal equaion Subac x an y fom boh sies o cancel he x an y. Reaange o pu he like ems ogehe. Subac. The x s an y s cancel on he ighhan sie of he equaion. The soluion o he lieal equaion is z A x y. Noice ha z is by iself on one sie of he equal sign, an ha hee ae no z s on he ohe sie of he equal sign. Pacice Poblem : Solve H c e fo. The soluion o his Pacice Poblem may be foun saing on page 10. Example 3 Solve a bc Z fo a. Soluion: We ae o solve fo a, which means we nee o ge he a by iself on one sie of he equal sign. Fis, ge he em wih he a by iself, by geing i of he bc em, which is subace fom he a. To ge i of subace quaniies in an equaion, you nee o a. You will no be one, hough: you will have o eal wih he muliplie, by iviing. This soluion will equie wo seps. a bc Z a bc bc Z bc a Z bc a Z bc a Z bc Oiginal equaion A bc o boh sies of he equaion o cancel he bc. Combine he like ems. The bc s cancel on he lefhan sie of he equaion. Divie boh sies by o cancel he ha muliplies he a. Cancel he s on he lef sie.
4 4 MAT 080: Lieal Equaions Z bc The soluion o he lieal equaion is a. Noice ha a is by iself on one sie of he equal sign, an ha hee ae no a s on he ohe sie of he equal sign. Thee ae wo ohe impoan hings o noe in his example: We canno o his poblem in one big sep. We neee o fis ge he em wih he a in i by iself, an hen eal wih is coefficien in a secon sep. When we ivie boh sies by, we ivie he enie igh sie of he equaion by. Don foge o o his. Pacice Poblem 3: Solve L W P fo W. The soluion o his Pacice Poblem may be foun saing on page 11. b Solving lieal equaions which equie facoing The ohe vaiey of lieal equaion ha we will consie ae hose which have wo ems conaining he unknown being solve fo. An example of such a poblem is: Solve ax bx c fo x. Wih an oinay equaion of his ype (fo example, 3x 5x 8) you pefom he moving all he xems o one sie sep o begin wih. You will o his wih a lieal equaion as well. A small poblem cops up, howeve. Look a he following wok, one in paallel beween he wo poblems: Equaion Lieal Equaion 3x 5x 8 oiginal poblem ax bx c 3x 5x 5x 5x 8 move he xem on he ax bx bx bx c igh o he ohe sie x 8 combine like ems ax bx c x 8 ivie o eliminae he??? muliplie faco, leaving jus x We have a poblem finishing he lieal equaion because ax an bx ae no like ems, an canno be combine. In he oinay equaion his coul be one
5 Objecive b: Solving lieal equaions which equie facoing 5 because 3x an 5x ae like ems. The ay will be save by facoing, which will un ax bx ino a muliplicaion poblem. Faco he common faco of x fom boh ems, which gives ax bx x(a b) Now we have somehing o ivie... he enie goup (a poblems we wee oing can now be coninue: b). The paallel Equaion Lieal Equaion x 8 combine /faco like ems x(a b) c x 8 ivie o eliminae he muliplie faco, leaving jus x x(a b) a b x 4 cancel x c a c a b b We sill use he same poceues o solve his kin of lieal equaion as we use o solve an oinay equaion, wih he aiion of a facoing sep. This sep will wok wheneve we have wo (o moe) ems wih he same vaiable being solve fo, which we ae unable o ohewise combine. Example 4 Solve 3x bx c fo x. Soluion: We ae o solve fo x, which means we nee o ge he x by iself on one sie of he equal sign. To ge o his poin we will nee o have all x ems on one sie of he equal sign. This is aleay he case: boh 3x an bx ae on he lef sie of he equaion, an no ems conaining x ae on he igh sie. Since we canno a he 3x an he bx we pefom he facoing sep escibe above. 3x bx c x(3 b) c x(3 b) 3 b x c 3 b c 3 b Faco he common faco of x fom boh ems. Divie boh sies by 3 b o cancel he 3 b ha muliplies he x. Cancel he 3 b on he lef sie of he equal sign.
6 6 MAT 080: Lieal Equaions c The soluion o he lieal equaion is x. Noice ha he x is by iself on 3 b one sie of he equal sign, an ha hee ae no x s on he ohe sie of he equal sign. Pacice Poblem 4: Solve ax 6 x D fo x. The soluion o his Pacice Poblem may be foun saing on page 11. Example 5 Solve a m fo. Soluion: We ae o solve fo, which means we nee o ge he by iself on one sie of he equal sign. To ge o his poin we will nee o have all ems on one sie of he equal sign. This is aleay he case: boh a an ae on he lef sie of he equaion, an no ems conaining ae on he igh sie. Since we canno a he a an he we pefom he facoing sep escibe above. The facoing sep nees a lile cae. a m (a 1) m (a 1) a 1 m a 1 m a 1 Faco he common faco of fom boh ems. Facoing fom will leave a 1! Don foge i! Divie boh sies by a 1 o cancel he a 1 ha muliplies he. Cancel he a 1 on he lef sie of he equal sign. m The soluion o he lieal equaion is. Noice ha he is by iself on a 1 one sie of he equal sign, an ha hee ae no s on he ohe sie of he equal sign. Pacice Poblem 5 Solve p p I fo p. The soluion o his Pacice Poblem may be foun saing on page 1. Example 6 Solve s s fo. Soluion: We ae o solve fo, which means we nee o ge he by iself on one sie of he equal sign. To ge o his poin we will nee o have all ems
7 Objecive b: Solving lieal equaions which equie facoing 7 on one sie of he equal sign. Thee ae ems wih s on each sie of he equal sign: an on he lef, an an s on he igh. We will fis nee o ge all he ems on he same sie of he equal sign. I will be moe efficien o move he o he igh sie. This can be one by subacing an fom boh sies. Then we will nee o o he facoing sep. s s s s s s Subac fom boh sies o move he em o he igh sie of he equaion. Combine like ems. The s cancel on he lefhan sie of he equaion. s ( s ) s ( s ) s s Faco he common faco of fom boh ems. Divie boh sies by s o cancel he s ha muliplies he. s Cancel he s on he igh sie of he equal sign. s s The soluion o he lieal equaion is. Noice ha he is by iself on one s sie of he equal sign, an ha hee ae no s on he ohe sie of he equal sign. Pacice Poblem 6 Solve a bc b fo b. The soluion o his Pacice Poblem may be foun saing on page 1. Example 7 Solve y x xy fo x. Soluion: We ae o solve fo x, an hee ae ems wih x s on each sie of he equal sign. We will fis nee o ge all he x ems on he same sie of he equal sign. I will be moe efficien o move he x o he igh sie. This can be one by aing x o boh sies. Then we will nee o o he facoing sep. The facoing sep equies a lile cae, an hee is a empaion ha nees o be fough a he en of he poblem.
8 8 MAT 080: Lieal Equaions y x xy y x x xy x A x o boh sies o move he igh sie of he equaion. x em o he y xy x Combine like ems. The x s cancel on he lefhan sie of he equaion. y x( y 1) y x( y 1) y 1 y 1 Faco he common faco of x fom boh ems. Facoing x fom x will leave a 1! Don foge i! Divie boh sies by y 1 o cancel he y 1 ha muliplies he x. y x Cancel he y 1 on he igh sie of he equal sign. y 1 y The soluion o he lieal equaion is x. Noice ha he x is by iself on y 1 one sie of he equal sign, an ha hee ae no x s on he ohe sie of he equal sign. WARNING! DO NOT CANCEL THE y s IN THE FRACTION! In his couse we will no lean he echniques fo simplifying facions wih unknowns in hem. Also, i woul be incoec o cancel he y s in his poblem egaless. So, when you ge facion answes in hese poblems, leave hem alone! Pacice Poblem 7 Solve 3c bc b fo b. The soluion o his Pacice Poblem may be foun saing on page 13.
9 Homewok poblems 9 Homewok Poblems Answes o Homewok poblems may be foun on page 15 a Solving lieal equaions which o no equie facoing Solve fo he vaiable inicae. 1. Solve fo I: IR E. Solve fo u: su 3. Solve fo j: Q i j k 4. Solve fo e: H c e 5. Solve fo c: f e 3 c 6. Solve fo y: x 7y Solve fo S: A Sw w 8. Solve fo c: bc 3a x b Solving lieal equaions which equie facoing Solve fo he vaiable inicae. 9. Solve fo x: 3x xy y 10. Solve fo m: am bm c 11. Solve fo k: G kp ak 1. Solve fo x: x x u 13. Solve fo y: ay y T 14. Solve fo w: A Sw w 15. Solve fo f: f g ef 16. Solve fo : 6 xy s 17. Solve fo h: A hc hb 18. Solve fo C: S C C 19. Solve fo b: bx 4x b 0. Solve fo g: 1 g gr
10 10 MAT 080: Lieal Equaions Soluions o Pacice Poblems Pacice Poblem 1: Solve fo. Soluion: To ge by iself I nee o ge i of he, which muliplies he. To ge i of muliplie quaniies in an equaion, you nee o ivie. Given lieal equaion Divie by o cancel he an isolae he. Cancel The soluion o he lieal equaion is. Noice ha is by iself on one sie of he equal sign, an ha hee ae no s on he ohe sie of he equal sign. Pacice Poblem : Solve H c e fo. Soluion: We ae o solve fo, which means you nee o ge he by iself on one sie of he equal sign. To ge he by iself, you nee o ge i of boh he c an he e, which ae ae o he. To ge i of ae quaniies in an equaion, you nee o subac. H c e H c e c e c e H c e c c e e H c e Given lieal equaion Subac c an e fom boh sies o cancel he c an e. Reaange o pu he like ems ogehe. Subac. The c s an e s cancel on he ighhan sie of he equaion. The soluion o he lieal equaion is H c e. Noice ha is by iself on one sie of he equal sign, an ha hee ae no s on he ohe sie of he equal sign.
11 Soluions o Pacice Poblems 11 Pacice Poblem 3 Solve L W P fo W. Soluion: We ae o solve fo W, which means we nee o ge he W by iself on one sie of he equal sign. To ge he em wih he W by iself, you fis nee o ge i of L em, which is ae o he W. To ge i of ae quaniies in an equaion, you nee o subac. You will no be one, hough: you will have o eal wih he muliplie, by iviing. This soluion will equie wo seps. L W P L L W P L Subac L fom boh sies of he equaion o cancel he L. W P L W P L W P L Combine he like ems. The L s cancel on he ighhan sie of he equaion. Divie boh sies by o cancel he ha muliplies he W. Cancel he s on he lef sie. The soluion o he lieal equaion is W P L. Noice ha W is by iself on one sie of he equal sign, an ha hee ae no W s on he ohe sie of he equal sign. Noe also ha we ivie he enie igh sie of he equaion by an ha he s o no cancel. Pacice Poblem 4 Solve ax 6 x D fo x. Soluion: We ae o solve fo x, which means we nee o ge he x by iself on one sie of he equal sign. To ge o his poin we will nee o have all x ems on one sie of he equal sign. This is aleay he case: boh ax an 6x ae on he lef sie of he equaion, an no ems conaining x ae on he igh sie. Since we canno a he ax an he 6x we pefom he facoing sep. ax 6x D x(a 6) D x(a 6) a 6 x D a 6 D a 6 Faco he common faco of x fom boh ems. Divie boh sies by a 6 o cancel he a 6 ha muliplies he x. Cancel he a 6 on he lef sie of he equal sign.
12 1 MAT 080: Lieal Equaions D The soluion o he lieal equaion is x. Noice ha he x is by iself on a 6 one sie of he equal sign, an ha hee ae no x s on he ohe sie of he equal sign. Pacice Poblem 5 Solve p p I fo p. Soluion: We ae o solve fo p, which means we nee o ge he p by iself on one sie of he equal sign. To ge o his poin we will nee o have all p ems on one sie of he equal sign. This is aleay he case: boh p an p ae on he lef sie of he equaion, an no ems conaining p ae on he igh sie. Since we canno a he p an he p we pefom he facoing sep escibe above. The facoing sep nees a lile cae. p p I p(1 ) I Faco he common faco of p fom boh ems. Facoing p fom p will leave a 1! Don foge i! p(1 ) I Divie boh sies by 1 o cancel he 1 ha 1 1 muliplies he p. I p 1 Cancel he 1 on he lef sie of he equal sign. I The soluion o he lieal equaion is p. Noice ha he p is by iself on 1 one sie of he equal sign, an ha hee ae no p s on he ohe sie of he equal sign. Pacice Poblem 6 Solve a bc b fo b. Soluion: We ae o solve fo b, which means we nee o ge he b by iself on one sie of he equal sign. To ge o his poin we will nee o have all b ems on one sie of he equal sign. Thee ae ems wih b s on each sie of he equal sign: a bc on he lef, an a b on he igh. We will fis nee o ge all he b ems on he same sie of he equal sign. I will be moe efficien o move he bc o he igh sie. This can be one by subacing a bc fom boh sies. Then we will nee o o he facoing sep.
13 Soluions o Pacice Poblems 13 a bc b a bc bc b bc a b bc Subac bc fom boh sies o move he b em o he igh sie of he equaion. Combine like ems. The bc s cancel on he lefhan sie of he equaion. a b( c ) a b( c) c c Faco he common faco of b fom boh ems. Divie boh sies by c o cancel he c ha muliplies he b. a b Cancel he c on he igh sie of he equal c sign. a The soluion o he lieal equaion is b. Noice ha he b is by iself on c one sie of he equal sign, an ha hee ae no b s on he ohe sie of he equal sign. Pacice Poblem 7 Solve 3c bc b fo b. Soluion: We ae o solve fo b, an hee ae ems wih b s on each sie of he equal sign. We will fis nee o ge all he b ems on he same sie of he equal sign. I will be moe efficien o move he bc o he igh sie. This can be one by subacing bc fom boh sies. Then we will nee o o he facoing sep. The facoing sep equies a lile cae, an hee is a empaion ha nees o be fough a he en of he poblem. 3c b bc 3c bc bc b bc 3c b bc 3 c b(1 c ) 3 c b(1 c) 1 c 1 c 3c 1 c b Subac bc fom boh sies o move he bc em o he igh sie of he equaion. Combine like ems. The bc s cancel on he lefhan sie of he equaion. Faco he common faco of b fom boh ems. Facoing b fom b will leave a 1! Don foge i! Divie boh sies by 1 c o cancel he 1 c ha muliplies he b. Cancel he 1 sign. c on he igh sie of he equal
14 14 MAT 080: Lieal Equaions 3c The soluion o he lieal equaion is b. Noice ha he b is by iself on 1 c one sie of he equal sign, an ha hee ae no b s on he ohe sie of he equal sign. WARNING! DO NOT CANCEL THE c s IN THE FRACTION! In his couse we will no lean he echniques fo simplifying facions wih unknowns in hem. Also, i woul be incoec o cancel he c s in his poblem egaless. So, when you ge facion answes in hese poblems, leave hem alone!
15 Answes o Homewok Poblems Answes o Homewok Poblems I E R. u s 3. j Q i k 4. e H c 5. c f e x y 7 o y x 7 7. S A w w 8. c x 3a b 9. x 3 y y 10. m c a b 11. k p G a 1. x u y T a w A S f g e 16. xy s h A b c 18. c s b 4x x 1 0. g 1 R 1
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