04. linear equation in one variable

8/12/2019 04. Linear Equation in One Variable

1/22

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2/22


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3/22


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4/22


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5/22


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6/22


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7/22


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:7 (1> x)

73x ; 5>(1> x) RKuitlpiyldm goth slnjs gy1:, thj IAK o` = edn :[

73x ; 5> 1> 5>x 73x + 5>x 5> 1> 1>x ; 5> 1>

1>

x1>;

1>

1>5>RNlvlnldm goth slnjs gy 1>[

x ; 5>Hjdaj, thj two perts erj 5> edn 1> 5> ; 73

Jx.: ^hj dukjretor o` e `reatlod ls > ijss thet thj

njdokldetor. L` 5 ls ennjn to goth lts dukjretoredn njdokldetor, lt gjaokjs 5/7. @ldn thj

`reatlod.

Qoi. Ijt thj njdokldetor o` thj `reatlod gj x. ^hjd,

Dukjretor o` thj `reatlod ; x >

@reatlod ;x

>x ...(l)

L ` 5 l s e nn jn t o g o t h l ts d uk jr et or e dn

njdokldetor, thj `reatlod gjaokjs7

5.

5x

5>x

;

7

5

5x

1x

;

7

5

7(x 1) ; x + 5 RZsldm aross-kuitlpilaetlod[ 7x 0 ; x + 5 7x x ; 0 + 5 x ; =Suttldm x ; = ld (l), wj mjt

@reatlod ;=

>= ;

=

1.

Hjdaj, thj mlvjd `reatlod ls=

1.

Jx.0 Qeuregh hes Ws 1> pelsj edn twjdty-`lvj pelsjaolds. L` thj dukgjr o` 7:-pelsj aolds gj twlaj

thj dukgjr o` :3-pelsj aolds, how kehy aolds o`

jeah bldn nojs hj hevj ?

Qoi. Ijt thj dukgjr o` :3-pelsj aolds gj x. ^hjd,

Dukgjr o` 7:-pelsj aolds ; 7x

Peiuj o` x `l`ty-pelsj aolds ; :3 x pelsj

; Ws533

x:3; Ws

7

x

Peiuj o` 7x twjdty-`lvj pelsj aolds ; 7: 7x pelsj

; Ws533

x:3; Ws

7

x

ôtei veiuj o` eii aolds ; Ws

7

x

7

x; Ws x

Gut, thj totei veiuj o` thj kodjy ls Ws 1>

x ; 1>

^hus, dukgjr o` :3-pelsj aolds ; 1>Dukgjr o` twjdty-`lvj pelsj aolds

; 7x ; 7 1> ; 08

Jx.= Ervldn hes Slmmy gedb. Lt ls ùii o` odj-rupjj

edn `l`ty-pelsj aolds. Lt aodtelds 1 tlkjs es kedy

`l ty pelsj aolds es odj rupjj aolds. ^hj totei

ekoudt o` thj kodjy ld thj gedb ls Ws 1:. How

kedy aolds o` jeah bldn erj thjrj ld thj gedb ?

Qoi. Ijt thjrj gj x odj rupjj aolds ld thj gedb. ^hjd,

Dukgjr o` :3-pelsj aolds ; 1x

Peiuj o` x odj rupjj aolds ; Ws x

Peiuj o` 1x `l`ty-pelsj aolds ; :3 1x pelsj

; 5:3 x ; pelsj ; Ws533

5:3x ; Ws

7

x1

ôtei veiuj o` eii thj aolds ; Ws

7

x1x

Gut, thj totei ekoudt o` thj kodjy ld thj ged ls

mlvjd es Ws 1:.

x +7

x1; 1:

7x + 1x ; =3 RKuitlpiyldm goth slnjs gy 7[

:x ; =3 :x: ;

:=3 x ; 5>

Dukgjr o` odj rupjj aolds ; 5>, Dukgjr o`:3 pelsj aolds ; 1x ; 1 5> ; >7.

Jx.8 Bedwer ls thrjj yjers oinjr thed Edlke. Qlx yjers

emo, Bedwers emj wes òur tlkjs Edlkes emj.

@ldn thj emjs o` Bdewer edn Edlke.

Qoi. Ijt Edlkes emj gj x yjers. ^hjd, Bedwers emj

ls (x + 1) yjers.

Qlx yjers emo, Edlkes emj wes (x 0) yjers

Lt ls mlvjd thet slx yjers emo Bedwers emj wes

òur tlkjs Edlkes emj. x 1 ; >(x 0)

x 1 ; >x 7>

x >x ; 7> + 1

1x ; 75

1

x1

;

1

75

x ; =

Hjdaj, Edlkes emj ; = yjers

Bedwers emj ; (x + 1) yjrs

; (= + 1) yjers ; 53 yjers.


9/22


Jx.< Hekln hes thrjj goxjs o` nl``jrjd rults. Gox E

wjlmhs 77

5bm korj thed Gox G edn Gox A wjlmhs

53>

5bm korj thed Gox G. ^hj totei wjlmht o` thj

goxjs ls >8>

1Hox kedy bm nojs Gox E wjlmh ?

Qoi. Qupposj thj gox G wjlmhts x bm.

Qlda gox E wjlmhs 77

5bm korj thed gox G edn

A wjlmhs 53>

5bm korj thed gox G.

]jlmht o` gox E ;

7

57x bm

;

7

:x bm ... (l)

]jlmht o` gox A ;

>

553x bm

;

>

>5x bm

^otei wjlmht o` eii thj goxjs

;

>

>5xx

7

:x bm

Gut, thj totei wjlmht o` thj goxjs ls mlvjd es

>8>

1bm ;

>

5

>5; >

5x + 53 + >x + >x + >5 ; 5[

57x + :5 ; 5

57

x57;

57

5>>

x ; 57Suttldm x ; 57 ld (l), wj mjt

]jlmht o` gox E ;

7

:57 bm ; 5>

7

5bm.

Jx.53 E ked soin ed ertlaei `or Ws >:3

Hjdaj, aost prlaj o` thj ertlaij ; Ws >:3.

Jx.55 :3 bm o` ed eiioy o` ijen edn tld aodtelds :3%

ijen. How kuah ijen kust gj kjitjn ldto lt to

kebj ed eiioy aodteldldm =:% ijen ?

Qoi. Ijed aodtjdts ld 533 bm eiioy ; 03 bm Ijen aodtjdts ld :3 bm eiioy

;

:3

533

03bm ; 13 bm

Ijt x bm o` ijen gj kjitjn ldto thj eiioy to kebj

lt ed eiioy aodteldldm =:% ijen. ^hjd,

]jlmht o` thj djw eiioy ; (:3 + x) bm

]jlmht o` ijen ld thj djw eiioy ; (13 + x) bm

Dow,

Ijen aodtjdts ld (:3 + x)bm eiioy ; (13 + x)bm

Ijen aodtjdts ld odj bm eiioy ;

x:3x13 bm

Ijen aodtjdts ld 533 bm eiioy

;

x:3

x13 533 bm

Sjrajdtemj o` ijen ld djw eiioy

;

533

x:3

x13%

Gut, thj pjrajdtemj o` ijen ld thj djw eiioy ls

mlvjd es =:%.

x:3

x13

533 ; =:

(13 + x) 533 ; =: (:3 + x)

RZsldm aross-kuitlpilaetlod[

1333 + 533x ; 1=:3 + =:x

533 x =:x ; 1=:3 1333

7:x ; =:3

7:

x7:;

7:

=:3

x ; 13

Hjdaj, rjqulrjn ijen to gj ennjn ; 13 bm.


10/22


Jx.57 ^hj suk o` two dukgjrs ls 7>55 @lrst dukgjr ; 5>55, Qjaodn dukgjr

; 7>55 ; 53=55 ;

5333

:. ^hjrj`orj, thj othjr dukgjr

wlii gj >: x.

Lt ls mlvjd thet thj retlo o` thj dukgjrs ls = 6 8.

x>:

x

;

8

=

8 x ; = (>: x) RGy aross-uitlpilaetlod[ 8x ; 15: =x 8x + =x ; 15: 5:x ; 15:

x ;5:

15:; 75

^hus, odj dukgjr ls 75 edn, Othjr dukgjr

; >: x ; >: 75 ; 7>

Ahjab Aijeriy, suk o` thj dukgjrs ; 75 + 7> ; >:, whlah

ls sekj es mlvjd ld thj progijk.

Wetlo o` thj dukgjrs ; 7>

75

; 8

=

, whlah ls sekj esmlvjd ld thj progijk.

^hus, our soiutlod ls aorrjat.

Jx.5> Nlvlnj Ws 5183 ekodm Ehkjn, Fohd edn Geglte

so thet thj ekoudt Ehkjn rjajlvjs ls : tlkjs e

skuah es Gegltes sherj edn ls 1 tlkjs es kuah

es Fohds sherj.

Qoi. Ijt Gegltes sherj gj Ws x. ^hjd,

Ehkjns sherj ; Ws :x

Fohds sherj ; ^otei ekoudt (Gegltessherj + Ehkjns sherj)

; WsR5183 (x + :x)[ ; Ws (5183 0x)Lt ls mlvjd thet Ehkjns sherj ls thrjj tlkjs

Fohds sherj.

:x ; 1(5183 0x) :x ; >5>3 58x :x + 58x ; >5>3 71x ; >5>3

x ;71

>5>3; 583

Gegltes sherj ; Ws 583, Ehkjns sherj; Ws(: 583) ; Ws ak. L` ijdmth edn grjenth erj jeah ldarjesjn gy

1 ak, thj erje o` thj djw rjatedmij wlii gj 85 ak7

korj thed thet o` thj mlvjd rjatedmij. @ldn thj

ijdmth edn grjenth o` thj mlvjd rjatedmij.

Qoi. Ijt thj grjenth o` thj mlvjd rjatedmij gj x ak.

^hjd, Ijdmth ; (x + >) ak

Erje ; Ijdmth Grjenth ; (x + >)x ; x7 + >x.]hjd ijdmth edn grjenth erj jeah ldarjesjn gy

1 ak.

Djw ijdmth ; (x + > + 1) ak ; (x + =) ak,

Djw grjenth ; (x + 1) ak

Erje o` djw rjatedmij ; Ijdmth Grjenth; (x + =) (x + 1)

; x(x + 1) + =(x + 1)

; x7 + 1x + =x + 75 ; x7 + 53x + 75

Lt ls mlvjd thet thj erj o` djw rjatedmij ls 85 ak7

korj thed thj mlvjd rjatedmij.

x7 + 53x + 75 ; x7 + >x + 85 x7 + 53x x7 >x ; 85 75 0x ; 03

x ;0

03; 53


11/22


^hus,

Ijdmth o` thj mlvjd rjatedmij

; (x + >)ak ; (53 + >) ak ; 5> ak

Grjenth o` thj mlvjd rjatedmij ; 53 ak

Ahjab Erje o` thj mlvjd rjatedmij ; (x7 + >x)ak7

; (537

+ > 53)ak7

; 5>3 ak7

Erje o` thj djw rjatedmij ; (x7 + 53x + 75)ak7

; (537 + 53 53 + 75)ak7 ; 775 ak7

Aijeriy, erje o` thj djw rjatedmij ls 85 ak7 korj

thed thet o` thj mlvjd rjatedmij, whlah ls thj

sekj es mlvjd ld thj progijk. Hjdaj, our edswjr

ls aorrjat.

Jx.50 Ed eitltutnj o` e trledmij ls lvj-thlrns thj ijdmth

o` lts aorrjspodnldm gesj. L` thj eitltunj wjrj

ldarjesjn gy > ak edn thj gesj gj njarjesjn gy

7 ak, thj erje o` thj trledmij wouin rjkeld thj

sekj. @ldn thj gesj edn thj eitltunj o` thj

trledmij.

Qoi. Ijt thj ijdmth o` thj gesj o` thj trledmij gj x ak.

^hjd,

Eitltunj ;

x

1

:ak ;

1

x:ak

Erje ;7

5(Gesj Eitltunj) ak7

;

1

x:x

7

5ak7 ;

0

x: 7

ak7

]hjd thj eitltunj ls ldarjesjn gy > ak edn thj

gesj ls njarjesjn gy 7 ak, wj hevj

Djw gesj ; (x 7) ak, Djw eitltunj ;

>

1

x:ak

Erje o` thj djw trledmij

;7

5(Gesj Eitltunj)

;7

5

)7x(>

1

x:ak7

;7

5

>

1

x:)7x( ak7

;75

)7x(>)7x(

1x: ak7

;7

5

8x>1

x53

1

x: 7

ak7

;7

5

>x7

1

x:

0

x: 7

ak7

Lt ls mlvjd thet thj erje o` thj mlvjd trledmij ls

sekj es thj erje o` thj djw trledmij.

0

x: 7

;0

x: 7

1

x:+ 7x >

0

x: 7

0

x: 7

+1x: 7x ; >

1

x: 7x ; > RKuitlpiyldm goth slnjs gy 1[

x ; 57 x ; 57akHjdaj, gesj o` thj trledmij ; 57 ak.

Eitltunj o` thj trledmij ;

57

1

:ak ; 73 ak

Ahjab ]j hevj,

Erje o` thj mlvjd trledmij ;

757

0

:ak ; 573ak7

Erje o` thj mlvjd trledmij

;

>57757

1

5:57

0

: 7ak ; 573ak7

^hjrj`orj, erje o` thj mlvjd trledmij ls thj sekj

es thet o` thj djw trledmij, whlah ls thj sekj es

mlvjd ld thj progijk. ^hus, our edswjr ls aorrjat.


12/22


U.5 E dukgjr ls es kuah mrjetjr thed 10 es ls ijss

thed 80. @ldn thj dukgjr.

U.7 @ldn e dukgjr suah thet whjd 5: ls sugtreatjn

`rok = tlkjs thj dukgjr, thj rjsuit ls 53 korj

thed twlxj thj dukgjr.

U.1 ^hj suk o` e retlodei dukgjr edn lts rjalproaei

ls0

51. @ldn thj dukgjr..

U.> ^hj suk o` two dukgjrs ls 58>. L` odj-thlrn o`

thj odj jxajjns odj-sjvjdth o` thj othjr gy 8,

`ldn thj skeiijr dukgjr.

U.: ^hj nl `jrjdaj o` two dukgjrs ls 55 edn odj-`l`th o` thjlr suk ls 7 edn thjlr pronuat

ls >1=, thjd `ldn thj egsoiutj nl``jrjdaj gjtwjjd

thj dukgjrs.

U.= ^hj suk o` two dukgjrs ls 5: edn thj suk o`

thjlr squerjs ls 551. @ldn thj dukgjrs.

U.8 ^hj evjremj o` òur aodsjautlvj jvjd dukgjrs

ls 7=. @ldn thj iermjst o` thjsj dukgjrs.

U.< ^hj suk o` thj squerjs o` thrjj aodsjautlvj ondukgjrs ls 7:15. @ldn thj dukgjrs.

U.53 O` two dukgjrs, > tlkjs thj skeiijr odj ls ijss

thed 1 tlkjs thj iermjr odj gy :. L` thj suk o`

thj dukgjrs ls iermjr thed : tlkjs thjlr nl``jrjdaj

gy 0, `ldn thj two dukgjrs.

U.55 ^hj retlo gjtwjjd e two-nlmlt dukgjr edn thj

suk o` thj nlmlts o` thet dukgjr ls > 6 5. L` thj

nlmlt ld thj udlts pieaj ls 1 korj thed thj nlmlt

ld thj tjds pieaj, whet ls thj dukgjr ?

U.57 E dukgjr aodslsts o` two nlmlts. ^hj suk o` thjnlmlts ls :3 ls nlvlnjn ldto two perts suah thet thj suk o`

thjlr rjalproaeis ls75

5. @ldn thj two perts.

U.5: L` thrjj dyukgjrs erj ennjn ld pelrs, thj suks

jquei 53, 5< edn 75. @ldn thj dukgjrs.

U.50 Wefjjvs emj e`tjr 5: yjers wlii gj : tlkjs hls emj

: yjers geab. ]het ls tj prjsjdt emj o` Wefjjv?

U.5= ^hj emjs o` two pjrsods nl``jrj gy 50 yjers. L`

0 yjers emo, thj jinjr odj gj 1 tlkjs es oin es

thj youdmjr odj, `ldn thjlr prjsjdt emejs.

U.58 ^hj pronuat o` thj emjs o` Edbjt edn Dlblte ls

7>3 . L` twlaj thj emj or Dlblte ls korj thed

Edblts emj gy > yjers, whet ls Dlbltes emj?

U.5< ^hj prjsjdt emj o` e èthjr ls 1 yjers korj thedthrjj tlkjs thj emj o` hls sod. ^hrjj yjers hjdaj,

èthjrs emj wlii gj 53 yjers korj thed twlaj thj

emj o` thj sod. @ldn thj prjsjdt emj o` thj èthjr.

U.73 Wohlt wes > tlkjs es oin es hls sod 8 yjers emo.

E`tjr 8 yjers, Wohlt wlii gj wtlaj es oin es hls

sod. ]het erj thjlr prjsjdt emjs ?

U.75 Odj yjer emo, thj retlo o` Meurevs edn Qeahlds

emj wes 0 6 = rjspjatlvjiy. @our yjers hjdaj, thls

retlo wouin gjaokj = 6 8. How oin ls Qeahld ?

U.77 Egheys emj e`tjr slx yjers wlii gj thrjj-sjvjdth

o` hls èthjrs emj et prjsjdt ?

EDQ]JWQ

5. 05 7. : 1. 7 /1 or 1/7 >.=7 :. 78 & 5= 0. >

=. = & 8 8.13 . 13 & 73 5:. 0, >, 5: 50. 53 yjers 5=. 5> & 13 yjers 58. 57 yjers

53 yjers 75. 10 yjers 77. :3 yjers


13/22


U.5 ^h j n l ``j rjd j g j tw jj d e d uk g jr e dn l t s

thrjj-`l`th ls :3. ]het ls thj dukgjr ?

(E) =: (G) 533(A) 57: (N) Dodj o` thjsj

U.7 L` e dukgjr ls njarjesjn gy > edn nlvlnjn gy 0,

thj rjsuit ls 8. ]het wouin gj thj rjsuit l` 7 ls

sugtreatjn `rok thj dukgjr edn thjd lt ls nlvlnjn

gy :?

(E) : (N) :>

U.> E dukgjr ls nougijn edn < ls ennjn. L` thj

rjsuitedt ls trjgijn, lt gjaokjs =:. ]het ls thet

dukgjr ?

(E) 1.: (G) 0

(A) 8 (N) Dodj o` thjsj

U.: ^hrjj-òurth o` e dukgjr ls 03 korj thed lts odj-

thlrn. ^hj dukgjr ls 6

(E) 8> (G) 538

(A) 5>> (N) Dodj o` thjsj

U.0 ]hjd 7> ls sugtreatjn `roke dukgjr, lt rjnuajs

to lts òur-sjvjdth. ]het ls thj suk o` thj nlmlts

o` thet dukgjr?

(E) 5 (G) , ls rjnuajn gy 75,

thj dukgjr ls 6

(E) 58 (G) 73 (A) 78 (N) 18

U.< E dukgjr whosj `l th pert ldarjesjn gy > ls

jquei to lts òurth pert nlkldlshjn gy 53, ls

(E) 7>3 (G) 703

(A) 7=3 (N) 783

U.53 ^hj nl` jrjdaj o` two dukgjrs ls 73% o` thj

iermjr dukgjr. L` thj skeiijr dukgjr ls 57, thj

iermjr odj ls 6

(E) 5: (G) 50

(A) 58 (N) 73

U.55 L` odj-sjvjdth o` e dukgjr jxajjns lts jijvjdth

pery gy 533, thjd thj dukgjr ls 6

(E) ==3 (G) 5533(A) 587: (N) 5 `rok thj pronuat, thj

dukgjr ls6

(E) 7 (G) 1 (A) > (N) 0

U.5> L` :3 ls sugtreatjn `rok two-thlrn o` e dukgjr,

thj rjsuit ls jquei to suk o` >3 edn odj-òurth

o` thet dukgjr. ]het ls thj dukgjr ?

(E) 5=> (G) 750 (A) 7>0 (N) 110

U.5: L` thj suk o` e dukgjr edn lts squerj ls 587,

whet ls thj dukgjr ?

(E) 5: (G) 70


U.50 ^wjdty tlkjs e posltlvj ldtjmjr ls ijss thed ltssquerj gy

(A) Aeddot gj njtjrkldjn

(N) Dodj o` thjsj

U.5= ^hrlaj thj squerj o` e deturei dukgjr njarjesjn

gy > tlkjs thj dukgjr ls jquei to :3 korj thed

thj dukgjr. ^hj dukgjr ls 6

(E) > (G) : (A) 0 (N) 53

U.58 ^hj suk o` e dukgjr edn lts rjalproaei ls odj-

jlmhth o` 1>. ]het ls thj pronuat o` thj dukgjredn lts squerj root ?

(E) 8 (G) 7=


U.5< ^wo-thlrn o` e posltlvj dukgjr edn750

7:o` lts

rjalproaei erj jquei. ^hj dukgjr ls 6

(E)57

:(G)

:

57

(A)5>>

7:(N)

7:

5>>

JXJWALQJ # 7


14/22


U.73 @ldn e posltlvj dukgjr whjd ldarjesjn gy 5= lsjquei to 03 tlkjs thj rjalproaei o` thj dukgjr.

(E) 1 (G) 53 (A) 5= (N) 73

U.75 E posltlvj dukgjr whjd njarjesjn gy > ls jquei

to 75 tlkjs thj rjalproaei o` thj dukgjr. ^hj

dukgjr ls6(E) 1 (G) : (A) = (N) 6 : 6 0 edn thjlr

evjremj ls 7:. ^hj iermjst dukgjr ls 6

(E) 13 (G) 17 (A) 10 (N) >7

U.70 ^hrjj dukgjrs erj ld thj retlo 1 6 > 6 0 edn thjlr

pronuat ls 5>. ^hj iermjst o` thjsj dukgjr ls 6

(E) 0 (G) 57


U.7= ^wo dukgjrs erj suah thet thj squerj o` odj ls

7>> ijss thed 8 tlkjs thj squerj o` thj othjr. L`

thj dukgjr gj ld retlo 1 6 > thj dukgjr erj6

(E) 0, 8 (G) 6 =. L` jeah ls ldarjesjn gy >, thj retlo

gjaokjs 1 6 :. ^hj iermjr dukgjr ls 6

(E) 10 (G) >8

(A) :0 (N) 0>

U.7< ^hj suk o` thrjj dukgjrs ls 70>. L` thj `lrst

dukgjr gj twlaj thj sjaodn edn thlrn dukgjr gj

odj-thlrn o` thj `lrst, thjd thj sjaodn dukgjr ls 6

(E) >8 (G) :>

(A) =7 (N) 8>

U.13 ^hj suk o` two dukgjrs ls 77. @lvj tlkjs odj

dukgjr ls jquei to 0 tlkjs thj othjr. ^hj glmmjr

o` thj two dukgjrs ls 6

(E) 53 (G) 57

(A) 5: (N) 50

U.15 Odj-`l`th o` e dukgjr ls jquei to8: o` edothjr

dukgjr. L` 1: ls ennjn to thj `lrst dukgjr, lt

gjaokjs `our tlkjs o` thj sjaodn dukgjr. ^hj

sjaodn dukgjr ls6

(E) 7: (G) >3

(A) =3 (N) 57:

U.17 ^hj suk o` two dukgjrs ls 7: edn thjlr nl``jrjdaj

ls 51, @ldn thjlr pronuat 6

(E) 53> (G) 55>

(A) 15: (N) 17:

U.11 L` thj suk o` two dukgjrs ls 11 edn thjlr

nl``jrjdaj ls 5:, thj skeiijr dukgjr ls 6

(E) < (G) 57

(A) 5: (N) 58

U.1> ^hj suk o` two dukgjrs ls >3 edn thjlr nl``jrjdaj

ls >. ^hj retlo o` thj dukgjrs ls 6

(E) 55 6 < (G) 55 6 58

(A) 75 6 5< (N) 77 6 yjers. L` thjlr

emjs erj ld thj rjspjatlvj retlo o` = 6


15/22


U.1= ^hj retlo gjtwjjd thj prjsjdt emjs o` S edn U

ls 0 6 =. L` U ls > yjers oin thed S, whet wlii gj

thj retlo o` thj emjs o` S edn U e`tjr > yjers?

(E) 1 6 > (G) 1 6 :

(A) > 6 1 (N) Dodj o` thjsj

U.18 ^hj retlo gjtwjjd thj prjsjdt emjs o` S edn U

ls : 6 = rjspjatlvjiy. L` thj nl``jrjdaj gjtwjjd Us

prjsjdt emj edn Ss emj e`tjr 0 yjers ls 7, whet

ls thj totei o` Ss edn Us prjsjdt emjs ?

(E) >8 yjers (G) :7 yjers

(A) :0 yjers (N) Dodj o` thjsj

U.1< Et prjsjdt, thj retlo gjtwjjd thj emjs o` Erud

edn Njjpeb ls > 6 1. E`tjr 0 yjers, Eruds emj wlii

gj 70 yjers. ]het ls thj emj o` Njjpeb et prjsjdt?

(E) 57 yjers (G) 5: yjers

(A) 53 Srjsjdt emjs o` X edn \ erj ld thj retlo : 6 0

rjspjatlvjiy. Qjvjd yjers hjdaj thls retlo wlii gj

gjaokj 0 6 = rjspjatlvjiy. ]het ls Xs prjsjdt

emj ld yjers ?

(E) 1: (G) >7

(A) >< (N) Aeddot gj njtjrkldn

U.>5 Srjsjdt emjs o` Qekjjr edn Ededn erj ld thj

retlo o` : 6 > rjspjatlvjiy. ^hrjj yjers hjdaj, thj

retlo o` thjlr emjs wlii gjaokj 55 6 < rjsjpjatlvjiy.

]het ls Ededns prjsjdt emj ld yjers?

(E) 7> (G) 7=

(A) >3 (N) Aeddot gj njtjrkldn

U.>7 Qlx yjers emo, thj retlo o` thj emjs o` Budei edn

Qemer wes 0 6 :. @our yjers hjdaj, thj retlo o`

thjlr emjs wlii gj 55 6 53. ]het ls Qemers emj et

prjsjdt ?

(E) 50 yjers (G) 58 yjers

(A) 73 yjers (N) Aeddot gj njtjrkldn

U.>1 ^hj totei o` thj emjs o` Feyedt, Srjk edn Qeredsh

ls . ]het ls thj prjsjdt emj o` Qeredsh?

(E) 7> yjers (G) 17 yjers

(A) 1> yjers (N) 18 yjers

U.>> ^hj retlo o` thj prjsjdt emjs o` two grothjrs ls

5 6 7 edn : yjers geab, thj retlo wes 5 6 1. ]het

wlii gj thj retlo o` thjlr emjs e`tjr : yjers ?

(E) 5 6 > (G) 7 6 1

(A) 1 6 : (N) : 6 0

U.>: Hltjsh ls >3 yjers oin edn Woddlj ls 03 yjers oin.

How kedy yjers emo wes thj retlo o` thjlr emjs

1 6 : ?

(E) : yjers (G) 53 yjers

(A) 73 yjers (N) 1= yjers

U.>0 ^hj retlo o` thj `ethjrs emj to hls sods emj ls

= 6 1. ^hj pronuat o` thjlr emjs ls =:0. ^hj retlo

o` thjlr emjs e`tjr 0 yjers wlii gj 6

(E) : 6 7 (G) 7 6 5

(A) 55 6 = (N) 51 6 = ^hj p rj sj dt e mj s o ` t hrjj p jrso ds e rj l d

proportlods > 6 = 6 : (N) Dodj o` thjsj

U.>8 ^hj retlo o` thj emjs o` e ked edn hls wl`j ls

> 6 1. E`tjr > yjers, thls retlo wlii gj < 6 =. L` et

thj tlkj o` kerrlemj, thj retlo wes : 6 1, thjd how

kedy yjers emo wjrj thjy kerrljn ?


(A) 57 yjers (N) 5: yjers

U.>< ^hj retlo gjtwjjd thj sahooi emjs o` Djjiek

edn Qheed ls : 6 0 rjspjatlvjiy. L` thj retlo

gjtwjjd thj odj-thlrn emj o` Djjiek edn hei` o`

Qheeds emj ls : 6 yjers emo edn Gs emj > yjers hjdaj 5 6 5.

]het ls thj retlo gjtwjjd Es emj > yjers hjdaj

edn Gs emj > yjers emo?

(E) 5 6 1 (G) 7 6 5

(A) 1 6 5 (N) > 6 5

U.:5 ^jd yjers emo, E wes hei` o` G ld emj. L` thj retlo

o` thjlr prjsjdt emjs ls 1 6 >, whet wlii gj thj

totei o` thjlr prjsjdt emjs ?


(A) >: yjers (N) Dodj o` thjsj

U.:7 E ls two yjers oinjr thed G who ls twlaj es oin

es A. L` thj totei o` thj emjs o` E, G edn A gj

7=, thjd how oin ls G ?

(E) = (G) 8

(A) < (N) 53


16/22


U.:1 E ked ls 7> yjers oinjr thed hls sod. Ld two

yjers, hls emj wlii gj twlaj thj emj o` hls sod.

^hj prjsjdt emj o` thj sod ls6

(E) 5> (G) 58 yjers

(A) 73 yjers (N) 77 yjers

U.:> Jlmhtjjd yjers emo, e èthjr wes thrjj tlkjs es

oin es hls sod. ^hjd thj suk o` thj prjsjdt emjs

o` thj sod edn thj èthjr ls6

(E) :> (G) =7

(A) 53: (N) 538

U.:: E pjrsods prjsjdt emj ls two-`l th o` thj emj o`

hls kothjr. E`tjr 8 yjers, hj wlii gj odj-hei` o`

thj emj o` hls kothjr. How oin ls thj kothjr et

prjsjdt?


(A) >3 yjers (N) >8 yjers

U.:0 êdyes mrednèthjr wes 8 tlkjs oinjr to hjr

50 yjers emo. Hj wouin gj 1 tlkjs o` hjr emj

8 yjers `rok dow. Jlmht yjers emo, whet wes thj

retlo o` êdyes emj to thet o` hjr mrednèthjr?

(E) 5 6 7

(G) 5 6 :

(A) 1 6 8

(N) Dodj o` thjsj

U.:= ^hj emj o` èthjr 53 yjers emo wes thrlaj thj emj

o` hls sod. ^jd yjers hjdaj, èthjrs emj wlii gj

twlaj thet o` hls sod. ^hj retlo o` thjlr prjsjdt

emjs ls 6

(E) :6 7 (G) = 6 1

(A) < 6 7 (N) 51 6 >

U.:8 @our yjers emo, thj èthjrs emj wes thrjj tlkjs

thj emj o` hls sod. ^hj totei o` thj emjs o` thj

èthjr edn thj sod e`tjr òur yjers, wlii gj

0> yjers. ]het ls thj èthjrs emj et prjsjdt?


(A) >> yjers (N) Dodj o` thjsj

U.:< Odj yjer emo, Sroklie wes òur tlkjs es oin es

hjr neumhtjr Qebshl. Qlx yjers hjdaj, Sroklie

emj wlii jxajjn hjr neumhtjr emj gy < yjer thj

retlo o` thj prjsjdt emjs o` Sroklie edn hjr

neumthjr ls6

(E) < 6 7 (G) 55 6 1

(A) 57 6 : (N) 51 6 >

U.03 ^hj suk o` thj prjsjdt emjs o` e èthjr edn hls

sod ls 03 yjers. Qlx yjers emo, èthjrs emj wes

`lvj tlkjs thj emj o` thj sod. E`tjr 0 yjers, sods

emj wlii gj6

(E) 57 yjers (G) 5> yjers

(A) 58 yjers (N) 73 yjers

EDQ]JW BJ\

U.Do 5 7 1 > : 0 = 8 < 53 55 57 51 5> 5: 50 5= 58 5< 73

Eds. A G N A A A E A N E N A A G N G G E E E

U.Do 75 77 71 7> 7: 70 7= 78 7< 13 15 17 11 1> 1: 10 1= 18 1< >3

Eds. A E A G E A E A A G G G E E E N N E G E

U.Do >5 >7 >1 >> >: >0 >= >8 >< :3 :5 :7 :1 :> :: :0 := :8 :< 03

Eds. E E N A G G G A N A N N N N A N G N N N


17/22


18/22


Qoi.5: Ijt thj dukgjrs gj x, y edn z. ^hjd,

x + y ; 53 ... (l)

y + z ; 5< ... (ll)

x + z ; 75 ... (lll)

Ennldm (l), (ll) edn (lll),

wj mjt 6 7(x + y + z) ; :3 or (x + y + z) ; 7:.

^hus, x ; (7: 52

z ; (7: 53) ; 5:

Hjdaj, thj rjqulrjn dukgjrs erj 0, >, edn 5:.

Qoi.50 Ijt Wefjjvs prjsjdt emj gj x yjers. ^hjd,

Wefjjvs e`j e`tjr 5: yjers ; (x + 5:) yjers.

Wefjjvs emj : yjers geab ; (x :) yjers.

x + 5: ; : (x :) x + 5: ; :x 7:

>x ; 53 x ; 53.

Hjdaj, Wefjjvs prjsjdt emj ; 53 yjers.

Qoi.5= Ijt thj emj o` thj youdmjr pjrsod gj x yjers.

^hjd, emj o` thj jinjr pjrsod ; (x + 50) yjers.

1(x 0) ; (x + 50 0) 1x 58 ; x + 53

7x ; 78 x ; 5>.

Hjdaj, thjlr prjsjdt emjs erj 5> yjers edn 13

yjers.

Qoi.58 Ijt Edblts emj gj x yjers. ^hjd, Dlbltes emj

; x

7>3yjers.

7 x

7>3 x ; > >83 x7 ; >x

x7 + >x >83 ; 3

(x + 7>) (x 73); 3 x ; 73.

Hjdaj, Dlbltes emj ;

73

7>3yjers ; 57 yjers.

Qoi.5< Ijt thj sods prjsjdt emj gj x yjers. ^hj, èthjrs

prjsjdt emj ; (1x + 1) yjers.

(1x + 1 + 1) ; 7 (x + 1) + 53

1x + 0 ; 7x + 50 x ; 53.Hjdaj, èthjrs prjsjdt emj ; (1x + 1)

; (1 53 + 1)yjers ; 11 yjers.

Qoi.73 Ijt sods emj 8 yjers emo gj x yjers. ^hjd,

Wohlts emj 8 yjers emo ; >x yjers.

Qods emj e tjr 8 yjers ; (x + 8) + 8

; (x + 50)yjers.

Wohlts emj e`tjr 8 yjers ; (>x + 8) + 8

; (>x + 50) yjers.

7 (x + 50) ; >x + 50 7x ; 50 x ; 8.Hjdaj, sods prjsjdt emj ; (x + 8) ; 50 yjers.

Wohlts prjsjdt emj ; (>x + 8) ; >3 yjers.

Qoi.75 Ijt Meurevs edn Qeahlds emjs odj yjers emo gj

0x edn =x yjers rjspjatlvjiy. ^hjd, Meurevs emj

> yjers hjdaj ; (0x + 5) + > ; (0x + :) yjers.

Qeahlds emj > yjers hjdaj ; (=x + 5) + > ; (=x

+ :) yjers.

:x=

:x0

;

8

= 8(0x + :) ; =(=x + :)

>8x + >3 ; >8

8x ; 0> x ; 8.

Hjdaj, Egheys èthjrs prjsjdt emj ; (:x + 53)

; :3 yjers.


19/22


Qoi.5 Ijt thj dukgjr gj x. ^hjd, x x:

1; :3

x ;

7

::3; 57:.

Qoi.7 Ijt thj dukgjr x. ^hjd,0

>x; 8x > ; >8

x > ; >8 x ; :7.

:

7x;

:

7:7;

:

:3; 53.

Qoi.1 Ijt thj dukgjr gj x. ^hjd,1

5o`

>

5o` x ; 5:

x ; 5: 57 ; 583.

Qo,rjqulrjn dukgjr ;

583

53

1; :>.

Qoi.> Ijt thj dukgjr gj x. ^hjd, 1(7x +

1x

1

5x ; 03

57

x:; 03 x ;

:

5703; 5>>.

Qoi.0 Ijt thj dukgjr gj x. ^hjd,

x 7> ;=

>x x

=

>x ; 7>

=

x ; 7>

x ;

1

=7>; :0.

Quk o` thj nlmlts ; (: + 0) ; 55.

Qoi.= Ijt thj dukgjr gj x. ^hjd, 5:x x ; 5x ; 5.

Qoi.8 Ijt thj dukgjr gj x. ^hjd,>

x; x 75

x ; >x 8> 1x ; 8> x ; 78.

Qoi.< Ijt thj dukjr gj x. ^hjd,

>x

:

5;

53x

>

5

73

x; 5> x ; 5> 73 ; 783.

Qoi.53 Ijt thj iermjr dukgjr gj x.

^hjd, x 57 ; 73% o` x x :x ; 57

:

x>; 57 x ;

>

:57; 5:.

Qoi.55 Ijt thj dukgjr gj x. ^hjd,=

5x

55

5x ; 533

==

x>;533 x ;

>

==33; 5

0x ; 7> x ; >.

Qoi.5> Ijt thj dukgjr gj x.

^hjd,1

7x :3 ;

>

5x + >3

1

7x

>

5x ; x ; x + :3

1x7 :x :3 ; 3 (1x + 53) (x :) ; 3 x ; :.

Hldts & Qoiutlod # 7


20/22


Qoi.58 Ijt thj dukgjr gj x. ^hjd, x +x

5;

8

1>

x

5x 7 ;

8

1> 8x7 1>x + 8 ; 3

>x

7

5=x + > ; 3

(>x 5) (x >) ; 3 x ; >.

.dodetureielsxes,

>

5xdjmijatldm

Wjqulrjn dukgjr ; > > ; > 7 ; 8.

Qoi.5< Ijt thj dukgjr gj x.

^hjd,1

7x ;

750

7:

x

5x7 ;

750

7:

7

1;

5>>

7:

x ;5>>

7:;

57

:.

Qoi.73 Ijt thj dukgjr gj x.

^hjd, x + 5= ;x

03 x7 + 5=x 03 ; 3

(x + 73) (x 1) ; 3 x ; 1.

Qoi.75 Ijt thj dukgjr gj x.

^hjd, x > ;x

75 x7 >x 75 ; 3

(x =) (x + 1) ; 3 x ; =.

Qoi.77 Ijt thj dukgjr gj x. ^hjd, x +

x

5; 1

x

5x

x

5x7 ; 1

x

5x7

x7 + 5 ; 1x7 1 7x7 ; > x7 ; 7

x ; 7 .

Qoi.71 Ijt thj dukgjr gj e edn g. ^hjd, eg ; 5=

e ; 5 edn g ; 5=.

Qo, 7e

5+ 7g

5; 77

77

ge

ge ; 7

77

)5=5(

)5=(5

;

78

; 7:

:

0

17

:x

7

:+ 1 ; 7:

:

0; 13

7

x

:=

7

x

01

x ;

01;

7

=; 1

7

5.

Qoi.7: Ijt t hj d uk gj rs g j > x, : x e dn 0 x. ^hj d,

1x0x:x> ; 7: :x ; 7: x ; :.

Iermjst dukgjr ; 0x ; 13.

Qoi.70 Ijt thj dukgjrs gj 1x, >x edn 0x.

^hjd, 1x >x 0x ; 5> =7x1 ; 5>

x1 ; 7= x ; 1.

Iermjst dukgjr 0x ; 58.

Qoi.7= Ijt thj dukgjrs gj 1x edn >x. ^hjd,

(>x)7 ; 8 (1x7) 77> 50x7 =7x7 77>

:0x

x7

; > x ; 7.Qo, thj dukgjrs erj 0 edn 8

Qoi.78 Ijt thj dukgjr gj >x edn =x.

^hjd,>x=

>x>

;

:

1

:(>x + >) ; 1 (=x + >) x ; 8.

Iermjr dukgjr ; =x ; :0.

Qoi.7< Ijt thj sjaodn dukgjr gj x. ^hjd, `lrst dukgjr

; 7x edn thlrn dukgjr ; 1

x7

7x + x +1

x7; 70>

1

x55; 70>

x ;

55

170>; =7.

Qoi.13 Ijt thj dukgjrs gj x edn (77 x).

^hjd, :x ; 0 (77 x) 55x ; 517 x ; 57.Qo, thj dukgjrs erj 57 edn 53.

Qoi.15 Ijt thj dukgjrs gj x edn y. ^hjd, x:

5; y

8

:

y ; x7:

8.

Dow, x + 1: ; >yx + 1: ; x7:

17 x

7:

=; 1:

x ;

=

7:1:; 57:.

Qjaodn dukgjr ; y ; x7:

8;

57:

7:

8; >3.


21/22


Qoi.17 Ijt thj dukgjrs gj x edn y. ^hjd, x + y ; 7:

edn x y ; 51.

>xy ; (x + y)7 (x y)7 ; (7:)7 (51)7

; 07: 50< ; >:0 xy ; 55>.

Qoi.11 Ijt thj dukgjrs gj x edn y.

^hjd, x + y ; 11 ....(l) edn x y ; 5: ....(ll)

Qoivldm (l) edn (ll), wj mjt 6 x ; 7> , y ;

>3; 53 (x + y) ; 53 (x y)

x ; >. Wjqulrjn retlo ; (0x + >) 6 (=x + >) ; 78 6 17

; = 6 8.

Qoi.18 Ijt thj prjsjdt emjs o` S edn U gj :x yjers edn

=x yjers rjspjatlvjiy.

^hjd, =x (:x + 0) ; 7 7x ; 8 x ; >. Wjqulrjn suk ; :x + =x ; 57x ; >8 yjers.

Qoi.1< Ijt thj prjsjdt emjs o` Erud edn Njjpeb gj >x

yjers edn 1x yjers rjspjatlvjiy.

^hjd, >x + 0 ; 70 >x ; 73 x ; :. Njjpebs emj ; 1x ; 5: yjers.

Qoi.>3 Ijt thj prjsjdt emjs o` X edn \ gj :x yjers edn

0x yjers rjspjatlvjiy.

^hjd,=x0

=x:

;

=

0 =(:x + =) ; 0 (0x + =)

x ; =. Xs prjsjdt emj ; :x ; 1: yjers.

Qoi.>5 Ijt thj prjsjdt emjs o` Qekjjr edn Ededn gj :x

yjers edn >x yjers rjspjatlvjiy.

^hjd,1x>

1x:

;

x ; 7> yjers.

Qoi.>7 Ijt thj emjs o` Budei edn Qemer 0 yjers emo gj

0x edn :x yjers rjspjatlvjiy.

^hjd,>)0x:(

>)0x0(

;

53

55 53(0x + 53)

; 55(:x +53) :x ; 53 x ; 7. Qemers prjsjdt emj ; (:x + 0) ; 50 yjers.

Qoi.>1 Ijt thj emjs o` Feyedt, Srjk edn Qeredsh 53

yjers emo gj 7x, 1x edn >x yjers rjspjatlvjiy.

^hjd, (7x + 53) + (1x + 53) + (>x + 53) ; > Ijt thj prjsjdt emjs o` thj two grothjrs gj x

yjers edn 7x yjers rjspjatlvjiy.

^hjd,:x7

:x;

1

51(x :) ; (7x ):x ;53.

Wjqulrjn retlo ; (x + :) 6 (7x + :) ; 5: 6 7:; 1 6 :

Qoi.>: Qupposj, thj retlo wes 1 6 :, x yjers emo.

^hjd,x03x>3

;15

:(>3 x) ; 1 (03 x)

7x ; 73 x ; 53.

Qoi.>0 Ijt thj prjsjdt emjs o` thj `ethjr edn sod gj =x

edn 1x yjers rjspjatlvjiy.

^hjd, =x 1x ; =:0 75x7 ; =:0 x7 ; 10 x ; 0.

Wjqulrjn retlo ; (=x + 0) 6 (1x + 0) ; >8 6 7>; 7 6 5

Qoi.>= Ijt thjlr prjsjdt emjs gj >x, =x edn x 8) + (=x 8) + ( ^hjlr prjsjdt emjs erj 50 yjers, 78 yjers edn10 yjers rjspjatlvjiy.

Qoi.>8 Ijt thj prjsjdt emjs o` thj ked edn hls wl j gj

>x edn 1x yjers rjspjatlvjiy.

^hjd,>x1

>x>

;

=

< =(>x + >) ; )

x ; 8.Qo, thjlr prjsjdt emjs erj 17 yjers edn 7> yjers

rjspjatlvjiy.


22/22


Qupposj thjy wjrj kerrljn z yjers emo.

^hjd,z7>

z17;

1

:; 1(17 z) ; :(7> z)

7z ; 7> z ; 57.

Qoi.>< Ijt thj sahooi emjs o` Djjiek edn Qheed gj :x

edn 0x yjers rjspjatlvjiy. ^hjd,

x07

5

x:1

5

;

x:

;

55 :x > ; 1x + >

7x ; 8 x ; >.Wjqulrjn retlo ; (:x + >)6 (1x >)

; 7> 6 8 ; 1 6 5.

Qoi.:5 Ijt thj emjs o` E edn G 53 yjers emo gj x edn

7x yjers rjspjatlvjiy.

^hjd,53x7

53x

;

>

1 >(x + 53) ; 1(7x + 53)

7x ; 53 x ; :. Quk o` thjlr prjsjdt emjs

; (x + 53) + (7x + 53) ; (1x + 73) ; 1: yjers.

Qoi.:7 Ijt As emj gj x yjers . ^hjd, Gs emj ; 7x yjers,

Es emj ; (7x + 7) yjers.

(7x + 7) + 7x + x ; 7= :x ; 7: x ; :.Hjdaj, Gs emj ; 7x ; 53 yjers.

Qoi.:1 Ijt thj sods prjsjdt emj gj x yjers. ^hjd, keds

prjsjdt emj ; (x + 7>) yjers.

(x + 7>) + 7 ; 7(x + 7) x + 70 ; 7x + > x ; 77.

Qoi.:> Ijt thj prjsjdt emjs o` thj èthjr edn sod gj 7x

edn x yjers rjspjatlvjiy.^hjd, (7x 58) ; 1(x 58) x ; 10. Wjqulrjn suk ; (7x + x) ; 1x ; 538 yjers.

Qoi.:: Ijt thj kothjrs prjsjdt emj gj x yjers.^hjd,

thj pjrsods prjsjdt emj ;

x

:

7yjers

8x

:

7;

7

5(x + 8)7(7x + >3) ; :(x + 8)

x ; >3.

Qoi.:0 50 yjers emo, ijt ^ ; x yjers edn M ; 8x yjers.

E`tjr 8 yjers `rok dow, ^ ; (x + 50 + 8) yjers edn

M ; (8x + 50 + 8) yjers.

8x + 7> ; 1 (x + 7>) :x ; >8.

8 yjers emo,M^

;8x8

8x ;

8:

>88

8:

>8

;>7>88

;:1

55.

Qoi.:= Ijt thj emjs o` èthjr edn sod 53 yjers emo gj

1x edn x yjers rjspjatlvjiy.

^hjd, (1x + 53) + 53 ; 7 R(x + 53) + 53[1x +73 ; 7x + >3 x ; 73.Wjqulrjn rtelo ; (1x + 53) 6 (x + 53)

=3 6 13 ; = 6 1.

Qoi.:8 Ijt thj emjs o` èthjr edn sod > yjers emo gj 1x

edn x yjers rjspjatlvjiy.

^hjd, R(1x + >) + >[ + R(x + >) + >[ >x ; >8 x ; 57.

@ethjrs prjsjdt emj ; 1x ; 10 yjers.Qoi.:< Ijt emjs o` Sroklie edn Qebshl 5 yjer emo gj >x

edn x yjers rjspjatlvjiy.

^hjd, R(>x + 5) + 0[ R(x + 5) + 0[ ; .

Qoi.03 Ijt thj prjsjdt emjs o` sod edn èthjr gj x edn(03 x) yjers rjspjatlvjiy.

^hjd, (03 x) 0 ; : (x 0) :> x; :x 13 0x ; 8> x ; 5>.

Qods emj e`tjr 0 yjer ; (x + 0) ; 73 yjers.

04. linear equation in one variable

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