Solve the following pairs of equations by reducing them to a pair of linear equations:
(i)
| (ii)
|
(iii)
| (iv)
|
(v)
| (vi)
|
(vii)
| (viii)
|
(i) Let 
and 
, then the equations becomes.
Using cross-multiplication method, we obtain,
p = 2 and q = 3
(ii) Putting 
in the given equations, we obtain
2p +3q = 2........(i)
4p - 9q = -1 .......(ii)
Multiplying equation (1) by 3,
we obtain 6p + 9q = 6....... (iii)
Adding equation (ii) and (iii), we obtain
10p = 5
Putting in equation (i), we obtain
= 
= 3q = 1
= 
p = 
= 
q = 
= 
Hence, x = 4 and y = 9.
(iii) Putting 
= 4p + 3y = 14
= 4p + 3y - 14 = 0.....................(i)
And, 3p - 4y = 23
= 3p - 4y -23 = 0.......................(ii)
By cross- multiplication , we get,
= 
= 
Now,
= 
= 
Also, p = 
Hence, x = 
(iv) Putting 
= 5p + q = 2....................(i)
= 6p - 3q = 1 .....................(ii)
Now, multiplying equation (i) by 3 we get,
= 15p + 3q = 6..............(iii)
Adding equations (ii) and (iii)
21 p = 7
= p = 
Putting value of p in equation (iii) we get,
= 
= 
= q = 
We know that,
p = 
= 3 = x - 1
= x = 4
And, q = 
= y - 2 = 3
= y = 5
Hence, x = 4 and y = 5
(v) 
= 
= 
= 
Putting 
7q - 2p = 5 ................(iii)
8q + 7p = 15 .................(iv)
multiplying equation (iii) by 7 and equation (iv) by 2 . we get,
49q - 14p = 35.................(v)
16q + 14p = 30..................(vi)
After adding equations (v) and (vi) . we get,
65q = 65
= q = 1
Putting value of q in equation (iv) , we get,
8 + 7p = 15
= 7p = 15 - 8 = 7
= p = 1
Now,
p = 
q = 
Hence , x = 1 and y = 1
(vi) 6x + 3y = 6xy
2x + 4y = 5xy
Putting 
6q + 3p - 6 = 0
2q + 4p - 5 = 0
By cross multiplication method , we get,
= 
= 
After comparing we get,
p = 1 and q = 
Now ,
p = 
Hence, x = 1 and y = 2
(vii) Putting 
10p + 2q - 4 = 0....................(i)
15p - 5q +2 = 0 ..................,,(ii)
By applying cross multiplication method , we get,
= 
= 
After comparing we get,
p = 
Now,
Adding equations (iii) and (iv) we get,
2x = 6
= x = 
Putting value of equation (iii) we get,
y = 2
Hence, x = 3 and y = 2
(viii) Putting 
p + q = 
p - q = 
Adding (i) and (ii) we get,
2p = 
= p = 
Putting value of p in (ii) we get,
= 
= q = 
Now,
p = 
q = 
Adding equations (iii) and (iv) we get,
6x = 6
= x = 1
Putting value of x in equation (iii) we get,
3(1) + y = 4
= y = 1
Hence, x = 1 and y = 1