kf7
10
10.0
/]vLo ;dLs/0f
(Linear Equation)
k'g/fjnf]sg
nt
Ce
nt
re
Ps hgf pkef]Stfn] 3 ls=u|f= sGo] Rofp / 2 ls=u|f= uf]a|] Rofpsf] d"No ?=1050 lt/]5g\ .
t/ csf{ pkef]Stfn] 2 ls=u|f= sGo] Rofp / 3 ls=u|f= uf]a|] Rofpsf] d"No ?=1200 lt/]5g\ eg]
k|To]s Rofpsf] d"No slt slt k¥of] xf]nf < sIffsf]7fdf 5nkmn tyf ts{af6 ;xL pTt/ kTtf
nufpg'xf];\ . o; ;d:ofnfO{ aLh ul0ftLo 9ª\un] xn ug{ ;lsG5 jf ;ls“b}g < ;lsG5 eg] o;nfO{
aLh ul0ftLo efiffdf s;/L k|:t't ug{ ;lsG5 < lzIfs;“u cfjZos k/fdz{ jf lgb]{zgdf n]Vg]
k|of; ug'{xf];\ .
lop
me
oxf“ dfly pNn]lvt b'O{ k|sf/sf Rofpsf] d"Nodf s]lGb|t eP/ b'O{ rno'St o'ukt /]vLo ;dLs/0fx¿
lgdf{0f / xn ug]{ tl/sf af/] x]/f+}“ .
3
ls=u|f= sGo] Rofp /
2
ls=u|f= uf]a|] Rofpsf] d"No
(ii)
2
ls=u|f= sGo] Rofp /
3
ls=u|f= uf]a|] Rofpsf] d"No
=
De
k|lt ls=u|f= sGo] Rofpsf] d"No
ve
(i)
?=
x
= ?.1050
=
?=1200
/ k|lt ls=u|f= uf]a|] Rofpsf] d"No= ?=
y
dfGbf
m
3x + 2y = 1050 … … … (i)
ulu
2x + 3y = 1200 … … … (ii)
Cu
r
ric
dflysf Ps hf]8L ;dLs/0fdf x / y rn x'g\ . oL b'j} k|yd l8u|Lsf] ;dLs/0f ePsfn] u|fkmdf
k|To]s ;dLs/0fn] ;/n /]vf lbG5g\ . oxf“ kTtf nufpg'kg]{ rnx¿ x / y sf dfgx¿ b'j}
;dLs/0fx¿nfO{ dfGo x'G5 . x / y sf] dfg kTtf nufpg'nfO{ g} ;dLs/0fsf] xn elgG5 . olb
b'j} ;dLs/0fx¿sf rnx¿nfO{ Pp6f lglZrt dfg dfq dfGo x'G5 eg] To:tf ;dLs/0fnfO{ o'ukt
/]vLo ;dLs/0f elgG5 . oxf“ o'ukt /]vLo ;dLs/0fdf b'O{ cf]6f rnx¿ ePsfn] tL ;dLs/0fnfO{
b'O{ rno'St o'ukt /]vLo ;dLs/0f klg elgG5 .
10.1
b'O{ rno'St /]vLo ;dLs/0f xn ug]{ ljlwx¿
(Methods of Solving Simultaneous Linear Equations )
df bf];|f] ;dLs/0faf6 x = y + 2 n]vL klxnf] ;dLs/0f x + y = 12 df
sf] dfg y + 2 /fVbf y sf] dfg s;/L kTtf nufpg ;lsG5, kTtf nufpg'xf];\ .
x + y = 12
96
/
x-y=2
ul0ft, sIff – (
x
o;sf ljlwx¿ lgDgfg';f/ 5g\ M
-s_ k|lt:yfkg ljlw
-v_ x6fpg] ljlw
-u_ n]vflrq ljlw
o;/L b'O{ jf ;f]eGbf a9L o¬ukt /]vLo ;dLs/0fdf Pp6f ;dLs/0faf6 Pp6f rnsf] dfg kTtf
nufP/ bf];|f] ;dLs/0fdf /fv]/ rnx¿sf] dfg kTtf nufpg] tl/sfnfO{ k|lt:yfkg ljlw elgG5 .
pbfx/0f
Ce
nt
re
k|lt:yfkg ljlw (Substitution Method)
10.1.1
1
xn ug'{xf];\ M
nt
3x + 2y = 1050 … … … (i)
3x = 1050 – 2y
lop
af6
x'G5 .
ve
(i)
cyjf,
x = 1050 − 2y
ca,
sf] dfgnfO{ ;dLs/0f
x
[b'j}
3
tk{m
3
n] efu ubf{]
De
oxf“ ;dLs/0f
me
2x + 3y = 1200 … … … (ii)
(ii)
df k|lt:yfkg ubf{
2100 − 4 y + 9y
= 1200
3
cyjf,
2100 + 5y = 3600
cyjf,
rri
[j|m;
5y = 3600 – 2100
[2100
cyjf,
y = 1500/5
[b'j}tkm{ 5

y
(oxf“, y
Cu
cyjf,
cyjf,
ca,
cu
lu
m
2(1050 − 2y)
+ 3y = 1200
3
u'0fg ubf{
]
nfO{ bfof“tkm{ nUbf ]
n] efu ubf{
]
y = 300
y = 300
x'G5 .
sf] dfg ;dLs/0f
sf] dfg ;dLs/0f
(ii)
(i) / (ii)
2x + 3 × 300 = 1200
ul0ft, sIff – (
df k|lt:yfkg ubf{
dWo] hxf“ k|lt:yfkg ubf{ klg dfGo x'G5 .)
x'G5 .
97
cyjf,
2x + 900 = 1200
cyjf,
2x = 1200 – 900
cyjf,
2x = 300
cyjf,
x=

x = 150
300
2
[ 900 nfO{
[ b'j}tkm{ 2
bfof“tkm{ nUbf]
n] efu ubf{ ]
x'G5 .
x
/
sf] dfg k|lt:yfkg u/L hf“r]/ x]/f}“ M
y
Ce
nt
re
oxf“ dfly xn ul/Psf
klxnf] ;dLs/0f
3x + 2y = 1050
cyjf,
bfof“ kIf
k'gM bf];|f] ;dLs/0f
2x + 3y = 1200
ve
=
lop
1050 = 1050
 afof“ kIf
2 × 150 + 3 × 300 = 1200
cyjf,
300 + 900 = 1200
cyjf,
1200 = 1200
De
cyjf,
=
ulu
afof“ kIf
bfof“ kIf
ric

me
450 + 600 = 1050
cyjf,

nt
3 × 150 + 2 × 300 = 1050
m
cyjf,
10.1.2 x6fpg]
Cu
r
t;y{ x / y sf k|To]s dfg b'j} ;dLs/0fdf dfGo 5g\ . xn u/]sf] aLh ul0ftLo k|lj|mof plrt 5 .
ljlw (Elimination Method)
Ps hgf cleefjsn] 4 cf]6f ;fgf / 3 cf]6f 7'nf sfkLsf] hDdf d"No ?= 100, csf]{ cleefjsn]
p:t} ;fgf 5 cf]6f / 7'nf 2 cf]6f sfkLsf] hDdf d"No ?= 90 ltg'{k5{ eg] ;f] k|lt PsfO ;fgf] /
7'nf] sfkLsf] dÒNo s;/L kTtf nufpg ;lsPnf x6fpg] ljlwåf/f xn ug'{xf];\ .
oxf“ ;fgf] sfkLsf] ahf/ d"No ?=
x
/ 7'nf] sfkLsf] ahf/ d"No ?=
y
dfGbf
4x + 3y = 100 … … … (i)
/
oxf“
98
5x + 2y = 90 … … … (ii)
y
nfO{ x6fP/
x
x'G5 .
dfg lgsfNg ;dLs/0f
(i)
nfO{
2
n] / ;dLs/0f
(ii)
nfO{
3
n] u'0fg ubf{
ul0ft, sIff – (
[4x + 3y = 100] × 2
/
[5x + 2y = 90] × 3
x'G5 .
cyjf,
8x + 6y = 200 … … … (iii)
15x + 6y = 270 … … … (iv)
oxf“ ;dLs/0f (iii) / (iv) df y sf] u'0ffª\s Pp6} cª\s
(iii) 36fp“bf y x6\5 / x dfq af“sL /xG5 .

6
Ce
nt
re
15x + 6y = 270 … … … (iv)
– (8x + 6y = 200) … … … (iii)
7x
[ 36fpbf
lrx\g abNg'k5{ lsg <]
[70 = 7 × 10 tyf b'j}tkm{ 7 n] efu ubf{
= 70
sf] dfg ;dLs/0f
(ii)
df k|lt:yfkg ubf{
5 × 10 + 2y = 90
cyjf,
2y = 90 – 50
cyjf,
y=
40
2
[90 – 50 = 40

y = 20
[40 = 2 × 20
nfO{ bfof“tkm{ nUbf
De
ve
[ 5 × 10 = 50
lop
x
af6 ;dLs/0f
]
me
 x = 10
ca
(iv)
nt
7  10
7
cyjf, x =
5 . To;}n] ;dLs/0f
/ b'j}tkm{
2
]
n] efu ubf{]
m
/ bfof“tkm{sf] c+z / x/af6
2
x6fp“bf]
cu
lu
ctM ;fgf] sfkLsf] d"No ?=10 k5{ eg] 7'nf] sfkLsf] d"No ?=20 kb{5 .
2
5nkmn ug'{xf];\ .
Cu
pbfx/0f
?
rri
s] o;nfO{ csf]{ ljlwaf6 klg ;dfwfg ug{ ;lsPnf
x6fpg] ljlwåf/f xn ug'{xf];\ / hf“r]/ x]g'{xf];\ M
3x + 2y = 11 … … … (i)
4x – 3y = 9 … … … (ii)
;dfwfg M oxf“ ;dLs/0f
u'0ffª\s ;dfg x'G5 .
ctM
(i)
nfO{
3
/ ;dLs/0f
(ii) nfO{ 2
n] u'0fg ubf{ b'j} ;dLs/0fdf
y
sf]
[3x + 2y = 11] × 3
[4x – 3y = 9] × 2
cyjf,
9x + 6y = 33 … … … (iii)
ul0ft, sIff – (
99
8x – 6y = 18 … … … (iv)
;dLs/0f
(iii) / (iv)
nfO{ hf]8\bf
y
x6\5 /
x
sf] dfg lg:sG5 .
 9x + 6y = 33
+ 8x – 6y = 18
ca,
= 51 [hf]8\bf
cyjf,
x=
lrx\g abNg' kb}{g lsg <]
51 3  17

17
17
 x = 3 [51 = 3 × 17
x
sf] dfg ;dLs/0f
(ii)
Ce
nt
re

17x
/ bfof“tkm{sf] c+z / x/af6
df k|lt:yfkg ubf{
3 = 3y

y=1
;dLs/0f
cyjf,
(i)
af6
3 × 3 + 2 × 1= 11
cyjf,
9 + 2= 11
cyjf,
11= 11
afof“ kIf
=
Cu
r
cyjf,
;dLs/0f
bfof“ kIf
k'gM ;dLs/0f
(ii)
af6
4x – 3y= 9
4×3–3×1= 9
cyjf,
12 – 3 = 9
cyjf,
9 = 9
afof“ kIf
=
lgisÈ{: t;y{
100
3x + 2y= 11
lop
/fvL hf“r]/ x]bf{
ve
y=1
De
/
m
x=3
me
cyjf,
ulu
ca,
12 – 3y = 9
]
ric

cyjf,
x6fp“bf
nt
4 × 3 – 3y = 9
17
bfof“ kIf
x
/
y
sf] dfg l7s 5 .
ul0ft, sIff – (
10.1.3 n]vflrq
ljlw (Graphical Method)
;dfGotM b'O{ rno'St /]vLo ;dLs/0fsf cfwf/df u|fkmdf n]vflrq cyjf ;/n /]vfx¿ lvRbf b'j}
;/n /]vfx¿ Ps cfk;df sfl6“bf aGg] k|ltR5]bg laGb' g} b'O{ rnx¿sf] dfg x'G5 . o;/L u|fkmdf
b'O{ cf]6f ;/n /]vfx¿ lvr]/ b'O{ cf]6f rnx¿sf] dfg lgsfNg] jf kTtf nufpg] ljlwnfO{ n]vflrq
ljlw elgG5 .
pbfx/0f
3
2y = 7 – x
2x  12
3

x  7
2
me
 Y
nt
3y = – 2x + 12
Ce
nt
re
n]vflrq ljlwåf/f xn ug'{xf];\ M 2x + 3y = 12 … … … (i) / x + 2y = 7 … … … (ii)
oxf“ ;dLs/0f (i) / (ii) nfO{ u|fkmdf k|:t't ug{ ;j{k|yd b'j} ;dLs/0fnfO{ y = mx + c sf] ¿kdf
¿kfGt/0f ug{'kb{5 .
h:t}M 2x + 3y = 12 /
x + 2y = 7
Y
(i)
3
2
/ ;dLs/0f
6
0
(ii)
sf] ;femf laGb'
x
y
(3, 2)
1
3
3
2
5
1
xf] .
Cu
rri
cu
lu
t;y{ ;dLs/0f
0
4
m
x
y
De
ve
lop
To;kl5 x sf ljleGg dfgx¿ /fvL y sf dfgx¿ lgsflnG5 . h;sf cfwf/df hf]8f ;dLs/0fnfO{
u|fkmdf k|:t't ul/G5 .
;dLs/0f (i) af6
;dLs/0f (ii) af6
;dLs/0f (i) sf j|mdhf]8f ;ª\Vofx¿ (0, 4), (3, 2) / (6, 0) tyf ;dLs/0f (ii) j|mdhf]8f ;ª\Vofx¿
(1, 3), (3, 2) / (5, 1) x'g\ . oL j|mdhf]8f ;ª\Vofx¿sf cfwf/df u|fkmdf ;/n /]vfx¿ lvRbf laGb'
(3, 2) df sfl6Psf 5g\ . sfl6Psf] pSt laGb'sf] j|mdhf]8f ;ª\Vof jf lgb]{zfª\s g} x / y sf xn
x'g\ . t;y{ x = 3 / y = 2 x'G5 .
ul0ft, sIff – (
101
cEof; 10
-s_
tn lbOPsf hf]8f /]vLo ;dLs/0fx¿nfO{ k|lt:yfkg ljlwåf/f xn u/L hf“Rg'xf];\ M
1.
x+y=7,
7.
10.
3x + y = 15
3x - 2y = 3,
x + 2y = 8
3x + 7y = 13 ,
9x + y = 19
3x + 4y = 27 ,
7x – 2y = 29
5.
8.
3x – 2y = 11,
x + 3y = 11
4x - 3y = 2,
3x + 4y = 39
4x – 3y = –1 ,
3x + 2y = 12
3.
6.
9.
Ce
nt
re
4.
2.
2x + 3y = 7,
3x + 5y = 24
5x – 2y = 20 ,
3x + 5y = 43
y = 5x + 1 ,
2x – 5y = –51
-v_
tn lbOPsf hf]8f ;dLs/0fx¿sf] xn x6fpg] ljlwåf/f xn u/L hf“Rg'xf];\ M
1.
7x + 9y = 41 ,
2x + 3y = 13
3x – 2y = 11,
x – 4y = –3
x – 6y + 36 = 0 ,
3x – 4y = 4
x + 2y = 7 ,
2x – y = 4
10.
8.
lop
5.
me
nt
3.
6.
9.
2x – y = 1
y = 3x
9x – 8y = 12 ,
2x + 3y = 17
2x + y – 12 = 0 ,
y = –12x + 6
ve
7.
5x + 6y = 27 ,
3x + 4y = 17
3x + y = 100
x – 4y = 120
3x + 2y – 340 = 0 ,
2x + 3y = 360
De
4.
2.
u|fkm ljlwsf] k|of]u u/L tn lbOPsf hf]8f /]vLo ;dLs/0fx¿sf] xn ug'{xf];\ M
1.
x + y = 7,
x- y=2
4x – 3y = 6 ,
3x – 4y = 1
2x + y = 6,
x + 2y = 6
3x + 2y = 8,
2x + y = 5
10.
ulu
5.
ric
7.
2.
Cu
r
4.
m
-u_
8.
3x + y = 15 ,
2x + 3y = 17
x + y = 25 ,
x–y=5
x + 2y = 10,
2x - y = 0
3.
6.
9.
3x – 2y = 4 ,
5x + y = 23
x + y = 9,
x + 2y = 12
5x + 2y = 165 ,
5x – 3y = 65
-3_
b'O{ ls=u|f= s'v/' fsf] df;' / 1 ls=u|f= v;Lsf] df;'sf] d"No ¿=1500 k5{ . olb 1 ls=u|f=
s'v'/fsf] df;' / 2 ls=u|f= v;Lsf] df;'sf] d"No ¿=1950 kb{5 eg] v;L / s'v'/fsf] df;'sf]
d"No k|lt ls=u|f= slt xf]nf < hf]8f /]vLo ;dLs/0fx¿ agfP/ xn ug'{xf];\ .
-ª_
s'g} ljb\ofnosf] v]n d}bfgsf] nDafOsf] Ps ltxfO rf}8fO 5 . olb pSt cfotfsf/ v]n
d}bfgsf] kl/ldlt 32 ld6/ eP ;f] v]n d}bfgsf] nDafO / rf}8fO kTtf nufpg'xf];\ .
102
ul0ft, sIff – (