Solve for x and y : 5 x + 6 y = 13 , 3 x + 4 y = 7 , x ≠ 0

The given equations are: 5 x + 6 y = 13 . . . i 3 x + 4 y = 7 . . . ii Putting 1 x = u , we get: 5 u + 6 y = 13 . . . iii 3 u + 4 y = 7 . . . iv On multiplying iii by 4 and iv by 6 , we get: 20 u + 24 y = 52 . . . v 18 u + 24 y = 42 . . . vi On subtracting vi from v , we get: 2 u = 10 u = 5 1 x = 5 ⇒ x = 1 5 On substituting x = 1 5 in i , we get: 5 1 / 5 + 6 y = 13 25 + 6 y = 13 6 y = 13 – 25 = - 12 y = - 2 Hence, the required solution is x = 1 5 and y = - 2 .

Solve for x and y : x + 6 y = 6 , 3 x - 8 y = 5 , y ≠ 0

The given equations are: x + 6 y = 6 . . . i 3 x - 8 y = 5 . . . i i Putting 1 y = v , we get: x + 6 v = 6 . . . iii 3 x – 8 v = 5 . . . iv On multiplying iii by 4 and iv by 3 , we get: 4 x + 24 v = 24 . . . v 9 x – 24 v = 15 . . . vi On adding v and vi , we get: 13 x = 39 x = 3 On substituting x = 3 in i , we get: 3 + 6 y = 6 6 y = 6 - 3 = 3 ⇒ 3 y = 6 ⇒ y = 2 Hence, the required solution is x = 3 and y = 2 .

Solve for x and y : 2 x - 3 y = 9 , 3 x + 7 y = 2 , y ≠ 0

The given equations are: 2 x - 3 y = 9 . . . i 3 x + 7 y = 2 . . . ii Putting, 1 y = v , we get 2 x - 3 v = 9 . . . iii 3 x + 7 v = 2 . . . iv On multiplying iii by 7 and iv by 3 , we get: 14 x - 21 v = 63 . . . . v 9 x + 21 v = 6 . . . vi On adding v and vi , we get: 23 x = 69 x = 3 On substituting x = 3 in i , we get: ( 2 × 3 ) - 3 y = 9 6 - 3 y = 3 ⇒ 3 y = - 3 ⇒ y = - 1 Hence, the required solution is x = 3 and y = - 1 .

Solve for x and y : 3 x - 1 y + 9 = 0 , 2 x + 3 y = 5 ( x ≠ 0 , y ≠ 0 )

Given, 3 x - 1 y + 9 = 0 , 3 x - 1 y = - 9 . . . . . . . i 2 x + 3 y = 5 . . . . . . . . i i Putting 1 x = u and 1 y = v , we get: 3 u – v = - 9 . . . . . i i i 2 u + 3 v = 5 . . . . . i v On multiplying i i i by 3 , we get: 9 u – 3 v = - 27 . . . . v On adding ( i v ) and ( v ) , we get: 11 u = - 22 ⇒ u = - 2 1 x = - 2 ⇒ x = - 1 2 On substituting x = - 1 2 in i , we get: 3 - 1 / 2 - 1 y = - 9 - 6 - 1 y = - 9 ⇒ 1 y = ( - 6 + 9 ) = 3 y = 1 3 Hence, the required solution is x = - 1 2 and y = 1 3

Solve for x and y : 9 x - 4 y = 8 , 13 x + 7 y = 101 , x ≠ 0 , y ≠ 0

The given equations are: 9 x - 4 y = 8 . . . i 13 x + 7 y = 101 . . . ii Putting 1 x = u and 1 y = v , we get: 9 u - 4 v = 8 . . . iii 13 u + 7 v = 101 . . iv On multiplying iii by 7 and iv by 4 , we get: 63 u - 28 v = 56 . . . v 52 u + 28 v = 404 . . . vi On adding v from vi , we get: 115 u = 460 u = 4 1 x = 4 ⇒ x = 1 4 On substituting x = 1 4 in i , we get: 9 1 / 4 - 4 y = 8 36 - 4 y = 8 ⇒ 4 y = 36 - 8 = 28 y = 4 28 = 1 7 Hence, the required solution is x = 1 4 and y = 1 7 .

Solve for x and y : 5 x - 3 y = 1 , 3 2 x + 2 3 y = 5 , ( x ≠ 0 and y ≠ 0 )

The given equations are: 5 x - 3 y = 1 . . . . . i 3 2 x + 2 3 y = 5 . . . . . . i i Putting 1 x = u and 1 y = v , we get: 5 u - 3 v = 1 . . . . . i i i 3 2 u + 2 3 v = 5 9 u + 4 v 6 = 5 9 u + 4 v = 30 . . . . i v On multiplying i i i by 4 and ( i v ) by 3 , we get: 20 u - 12 v = 4 . . . . . . . v 27 u + 12 v = 90 . . . . . . . . v i On adding ( i v ) and ( v ) , we get: 47 u = 94 ⇒ u = 2 1 x = 2 ⇒ x = 1 2 On substituting x = 1 2 in i , we get: 5 1 / 2 - 3 y = 1 10 - 3 y = 1 ⇒ 3 y = ( 10 - 1 ) = 9 y = 3 9 = 1 3 Hence, the required solution is x = 1 2 and y = 1 3 .