8g - MwYZ
c„ôv 1
Aa¨vq- 6 ( mij mnmgxKiY)
Abykxjbx- 6.1
1bs m„Rbkxj cÖkœt
x y 2a
ax by a 2 b 2
K. cÖ`Ë mgxKiY؇qi mgvavb x 3a, y a wKbv hvPvB Ki|
L. mgxKiYØq‡K cÖwZ¯’vcb c×wZ‡Z mgvavb Ki|
M. cÖ`Ë mgxKiYØq‡K Acbqb c×wZ‡Z mgvavbKi|
mgvavb
K. cÖ`Ë mgxKiYØq,
x y 2a
ax by a 2 b 2
x 3a, y a n‡j
1g mgxKi‡Yi evgcÿ= 3a a
= 2a
evgcÿ= Wvb cÿ
Avevi,
2q mgxKi‡Yi evgcÿ = a,3a b.a
= 3a 2 ab
evgcÿ Wvb cÿ
myZivs x 3a Ges y a cÖ`Ë mgxKi‡Yi gyj bq|
L. cÖ`Ë mgxKiY
x y 2a. ... ... ...(i )
ax by a 2 b 2 ... ... ...(ii )
mgxKiY ii n‡Z cvB
y 2a x
ev, y x 2a ... ... ...(iii )
[Dfq cÿ‡K Øviv 1 ¸Y K‡i]
mgxKiY iii n‡Z cÖvß y Gi gvb mgxKiY ii G ewm‡q cvB
ax b( x 2a) a 2 b 2
ev, ax bx 2ab a 2 b 2
ev, ax bx 2ab a 2 b 2
ev, ax bx a 2 b 2 2ab
8g - MwYZ
ev, x(a b) (a b) 2
c„ôv 2
2
a b
ev, x
ab
x a b
x Gi gvb mgxKiY iii G ewm‡q cvB,
y a b 2a
y ba
wb‡b©q mgvavb, x, y a b, b a
M. cÖ`Ë mgxKiYØq
x y 2a ... ... ...(i)
ax by a 2 b 2 ... ... ...(ii )
mgxKiY i †K b Øviv Ges mgxKiY ii †K 1 Øviv ¸Y K‡i cvB,
bx by 2ab
ax by a 2 b 2
†hvM K‡i, ax bx a 2 b 2 2ab
ev, a bx a 2 2ab b 2
ev,
2
a b
x
ab
a ba b
a b
x Gi gvb mgxKiY i -G ewm‡q cvB
ev, x
a b y 2a
ev, a b 2a y
ev, b a y
y ba
wb‡Y©q mgvavbt x, y a b, b a
2bs m„Rbkxj cÖkœt
2 x 3 y 7 Ges 6x 7 x 5 `ywU exRMwYZxq ivwk
K. mgxKiY `ywU‡Z KhwU K‡i PjK Av‡Q?
L. mgxKiYØq‡K Acbqb c×wZ‡Z mgvavb Ki|
M. cÖvß mgvav‡bi ïw× cixÿv Ki|
mgvavb
8g - MwYZ
K. 1g mgxKiY 2 x 3 y 7 G PjK x I x
2q mgxKiY 6 x 7 y 5 G PjK x I x
mgxKiY `ywU‡Z 2wU K‡i PjK Av‡Q|
L. cÖ`Ë mgxKiY
2 x 3 y 7 ... ... ...(i )
6 x 7 y 5 ... ... ...(ii )
mgxKiY i †K 7 Øviv Ges mgxKiY ii †K 3 Øviv ¸Y K‡i cvB
14 x 21 y 49
18 x 21y 15
‡hvM K‡i, 32 x
ev, x
64
64
32
x 2
x Gi gvb mgxKiY i G ewm‡q cvB
2.2 3 y 7
ev, 4 3 y 7
ev, 3 y 7 4
ev, 3 y 3
ev, y
3
3
y 1
wb‡Y©q mgvavb x, y 2.1
M. cÖ`Ë mgxKiY
2x 3y 7
6x 7 y 5
L ‡_‡K cvB
x 2, y 1
x 2 Ges y 1 n‡j 1g mgxKi‡Yi
evgcÿ= 2 2 3 1
43
7
evgcÿ= Wvb cÿ
Avevi,
x 2 Ges y 1 n‡j 2q mgxKi‡Yi
evgcÿ= 6 2 7 1
c„ôv 3
8g - MwYZ
12 7
c„ôv 4
5
evgcÿ= Wvb cÿ
cÖvß mgvavb mwVK|
AwZwi³ cÖ¨vKwUm kxU
1. `yBwU mgxKiY j¶ Ki :
K. 1g mgxKiY †_‡K y Gi gvb Gi mvnv‡h¨ cÖKvk Ki|
L. cÖwZ¯’vcb c×wZ‡Z mgvavb K‡i x I y Gi gvb †ei Ki|
M. ÔLÕ Gi gvb e¨envi K‡i Acbqb c×wZ‡Z ax – by = 11 ax + by =19Ges mgxKiYØq
†_‡K a I bGi gvb †ei Ki|
2. mgxKiY `yBwU wb¤œi~c :
x
b
y ax – by = c
a
a
K. 1g mgxKiY‡K mij mgxKiY AvKv‡i †jL|
L. Acbqb c×wZ‡Z mgvavb K‡i x I y Gi gvb wbY©q Ki|
M. cÖvß mgvavb mgxKiYØq‡K wm× K‡i wK bv hvvB Ki|
3. wb‡i mgxKiY `yBwU j¶ Ki : 3x – 2y = 0 .... (i)
17x – 7y = 13...(ii)
K. (1,1) we›`ywU (i) bs mgxKi‡Yi mgvavb wKbv hvvB Ki|
L. mgxKiY †RvUwU Acbqb c×wZ‡Z mgvavb Ki|
4
M. ‡jLw‡Îi mvnv‡h¨ mgxKiY `yBwU‡K mgvavb K‡i ÔLÕ As‡k cÖvß djvdj hvvB Ki|
4. wb‡i mgxKiY †RvUwU j¶ Ki : 2x + y = k ......(i)
x – y = 2 ......(ii)
K. mgxKiY (ii) n‡Z y A‡¶i Dci Aew¯’Z Ggb GKwU we›`y wbY©q Ki|
L. mgxKiY (ii) Gi †Kvb we›`y‡Z fzR †KvwUi wØMyY n‡e?
8g - MwYZ
c„ôv 5
M. k = 7 n‡j mgxKiY †RvUwU mgvavb Ki|
5. wb‡i mgxKiY †RvUwU j¶ Ki : x + 4y = 14......(1)
7x – 3y = 5 .......(2)
K. mgxKiY (1) y A¶‡K †Kvb we›`y‡Z †Q` K‡i?
2
L. mgxKiY †RvUwU †h‡Kv‡bv cw×wZ‡Z mgvavb Ki|
4
M. (1) I (2) mgxKi‡Yi aªye c‡`i ¯’‡j h_vKª‡g 1 I 38 _vK‡j mgvavb wKiyc n‡e?
6.
3 x y 2
2
5
3
2
( x 1) y 1
1
3
4
GKwU `yB jKwewkó mgxKiY †RvU|
K. mgxKiY †RvU‡K miji~‡c cÖKvk Ki|
L. cÖwZ¯’vcb c×wZ‡Z mgvavb Ki|
M. ‡jLw‡Îi gva¨‡g mgvavb K‡i mZ¨Zv hvvB Ki|
7. 25x +27y = 131
27x + 25y = 129
GKwU `yB vjK wewkó mgxKiY †RvU|
K. mgxKiYMy‡jv‡K y Gi mgvb mnM wewkó mgxKi‡Y
cÖKvk Ki|
L. Acbqb c×wZ‡Z mgvavb Ki|
M. mgvav‡bi mZ¨Zv hvvB Ki Ges mgvavb we›`ywU †jL KvM‡R ¯’vcb Ki|
8. wb‡i mgxKiY †RvUwU j¶ Ki :
x + ay = b ......... (i)
ax – by = c ........(ii)
K. (b, o)we›`ywU †Kvb mgxKi‡Yi GKwU g~j?
L. a = 1, b = 2 Ges 3 n‡j mgxKiY †RvUwU mgvavb Ki|
M. Acbqb c×wZ‡Z cÖ`Ë mgxKiY †RvUwU mgvavb Ki|
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