Q. 74.0( 116 Votes )

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) 152 x - 378 y = - 74

-378 x + 152 y = - 604

Answer :

(i) p x + q y = p - q … (1)
    q x - p y = p + q … (2)


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


we obtain px + pq y = p2 - pq … (3)


qx - pq y = pq + q2 … (4)


Adding equations (3) and (4),


we obtain p2x + q2 x = p2 + q2


(p2 + q2) x = p2 + q2



From equation putting the value of x (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 ax + ab y = ac … (3)


bx + ab y = b + bc … (4)


Subtracting equation (4) from equation (3),


(a2 - b2) x = ac - bcb



From equation (1), we obtain ax + by = c, now putting the value of x in the equation









(iii)


Or, bx - ay = 0 … (1)


ax + by = a2 + b2 … (2)


Multiplying equation (1) and (2) by b and a respectively, we obtain bx - ab y = 0 … (3)


ax + ab y = a3 + ab2 … (4)


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


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


By using (1), and putting the value of x in the equation 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), and putting the value of x in the equation we obtain


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


(a + b) y = - 2ab



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


-378 x + 152 y = - 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


-119780 y = -119780


y = 1


put the value in eq 1,
152 x - 378 x 1 = - 74
152 x = 378 - 74
152 x = 304
x = 2
we get x = 2.

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