(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