Solve the following pair of linear equations.


(i) px + qy = p - q


qx - py = p + q


(ii) ax + by = c


bx + ay = 1 + c


(iii) 


ax + by = a2 + b2


(iv) (a - b) x + (a + b) y = a2 - 2ab - b2


(a + b) (x + y) = a2 + b2


(v) 152x - 378y = - 74


-378x + 152y = - 604


(i) px + qy = p - q … (1) qx - py = p + q … (2)


 


Multiplying equation (1) by p and equation (2) by q,


 


we obtain p2x + pqy = p2 - pq … (3)


 


q2x - pqy = pq + q2 … (4)


 


Adding equations (3) and (4),


 


we obtain p2x + q2 x = p2 + q2


 


(p2 + q2) x = p2 + q2


 



 


From equation (1),


 


we obtain p (1) + qy = p - q


 


qy = - q


y = - 1


 


(ii) ax + by = c … (1) bx + ay = 1 + c … (2)


 


Multiplying equation (1) by a and equation (2) by b,


 


we obtain a2x + aby = ac … (3)


 


b2x + aby = b + bc … (4)


 


Subtracting equation (4) from equation (3),


 


(a2 - b2) x = ac - bcb


 



 


From equation (1), we obtain ax + by = c


 



 



 



 



 



 



 



 


 


 


(iii)


 


Or, bx - ay = 0 … (1)


 


ax + by = a2 + b2 … (2)


 


Multiplying equation (1) and (2) by b and a respectively, we obtain b2x - aby = 0 … (3)


 


a2x + aby = a3 + ab2 … (4)


 


Adding equations (3) and (4), we obtain b2x + a2x = a3 + ab2


 


x (b2 + a2) = a (a2 + b2) x = a


 


By using (1), we obtain b (a) - ay = 0


 


ab - ay = 0


ay = ab


 


y = b


 


 


 


(iv) (a - b) x + (a + b) y = a2 - 2ab - b2 … (1)


 


(a + b) (x + y) = a2 + b2


 


(a + b) x + (a + b) y = a2 + b2 … (2)


 


Subtracting equation (2) from (1),


 


we obtain


 


(a - b) x - (a + b) x = (a2 - 2ab - b2) - (a2 + b2) (a - b - a - b) x = - 2ab - 2b2


 


- 2bx = - 2b (a + b) x = a + b


 


Using equation (1), we obtain


 


(a - b) (a + b) + (a + b) y = a2 - 2ab - b2a2 - b2 + (a + b) y = a2 - 2ab - b2


 


(a + b) y = - 2ab


 



 


(v) 152x - 378y = - 74 ------------(1)


 


-378x + 152y = -604 -------- (2)


 


Multiply eq (2) by 152 and equation (1) by 378


 


378 × 152x – 3782y = -74 × 378


 


-378 × 152x + 1522y = -604 × 152


 


Adding both the questions we get


 


(1522 – 3782)y = -119780


 


-119780y = -119780


 


y = 1


 


put the value in eq 1, we get x = 2

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