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


1
1