 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 
ab
x  a  b
x Gi gvb mgxKiY iii  G ewm‡q cvB,
y  a  b  2a
y ba
 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  bx  a 2  2ab  b 2
ev,
2

a  b
x
ab
a  ba  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 ba
 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
 43
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|