Solve for x and y : 35 x + y + 14 x − y = 19 , 14 x + y + 35 x − y = 37

35 x + y + 14 x − y = 19 , 14 x + y + 35 x − y = 37 Put 1 x + y = u and 1 x - y = v So, we get 35 u + 14 v = 19 . . . . . . . . . ( i ) and 14 u + 35 v = 37 . . . . . . . . . . ( i i ) Adding i and i i , we get 49 u + 49 v = 56 ⇒ 7 u + 7 v = 8 ⇒ 14 u + 14 v = 16 . . . ( i i i ) Subtract ( i i i ) from ( i ) , we get 7 u = 1 ⇒ u = 1 7 Substituting u = 1 7 in i i i , we get v = 1 So, 1 x + y = 1 7 . . . . . . . . . ( i v ) and 1 x − y = 1 . . . . . . . . . ( v ) Adding ( i v ) and v , we get: 2 x = 8 ⇒ x = 4 Substituting x = 4 in i v , we get y = 3

If 1 is added to both the numerator and the denominator of a fraction, it becomes 4 5 , If, however, 5 is subtracted from both the numerator and the denominator, the fraction becomes 1 2 . Find the fraction.

Let the fraction be x y According to the first condition, x + 1 y + 1 = 4 5 ⇒ 5 x + 5 = 4 y + 4 ⇒ 5 x − 4 y = − 1 . . . ( i ) According to the second condition, x − 5 y − 5 = 1 2 ⇒ 2 x − 10 = y − 5 ⇒ 2 x − y = 5 . . . ( i i ) Multiply ( i i ) by 4 and subtract it from i , we get, - 3 x = - 21 ⇒ x = 7 Substituting x = 7 in i , we get y = 9 . So, the fraction is 7 9

Solve: a x b − b y a = a + b , a x − b y = 2 a b

a x b − b y a = a + b a 2 x + b 2 y = a 2 b + a b 2 . . . ( i ) a x - b y = 2 a b . . . ( i i ) Multiply i i by - b , so we get - a b x + b 2 y = - 2 a b 2 . . . ( i i i ) Adding i and i i i , we get a 2 x − a b x = a 2 b + a b 2 − 2 a b 2 ⇒ x ( a 2 − a b ) = a 2 b − a b 2 ⇒ x = a b a - b a a - b ⇒ x = b Substituting x = b in i i , we get a b - b y = 2 a b ⇒ - b y = a b ⇒ y = - a So, x = b and y = - a