Solve the following simultaneous equations. 2 x - 3 y = 15 ; 8 x + 5 y = 77

Given, 2 x - 3 y = 15 ; 8 x + 5 y = 77 Let 1 x = u and 1 y = v So, the equation becomes 2 u - 3 v = 15 ...............(i) 8 u + 5 v = 77 .............(ii) Multiply (i) with 4 we get 8 u - 12 v = 60 ...........(iii) Subtracting (ii) from (iii) 8 u + 5 v = 77 - 8 u + 12 v = 60 17 v = 17 ⇒ 17 v = 17 ⇒ v = 1 Putting the value of v in (i) = 2 u - 3 ( 1 ) = 15 ⇒ 2 u = 15 + 3 = 18 ⇒ u = 9 Thus, 1 x = u = 9 ⇒ x = 1 9 1 y = v = 1 ⇒ y = 1 So, ( x , y ) = 1 9 , 1

Solve the following simultaneous equations. 10 x + y + 2 x - y = 4 ; 15 x + y - 5 x - y = - 2

Given, 10 x + y + 2 x - y = 4 ; 15 x + y - 5 x - y = - 2 Let 1 x + y = u and 1 x - y = v So, the equation becomes 10 u + 2 v = 4 .............(i) 15 u - 5 v = - 2 .........(ii) Multiplying (i) with 5 and (ii) with 2 Then, we will get 50 u + 10 v = 20 .........(iii) 30 u - 10 v = - 4 ........(iv) Adding (iii) and (iv) we get = 80 u = 16 ⇒ u = 16 80 = 1 5 Putting this value in (I) 10 × 1 5 + 2 v = 4 ⇒ 2 + 2 v = 4 ⇒ v = 1 1 x + y = 1 5 and 1 x - y = 1 ⇒ x + y = 5 and x - y = 1 Solving these equations we get x = 3 and y = 2

Solve the following simultaneous equations 27 x - 2 + 31 y + 3 = 85 ; 31 x - 2 + 27 y + 3 = 89

Given, 27 x - 2 + 31 y + 3 = 85 ; 31 x - 2 + 27 y + 3 = 89 Let 1 x - 2 = u and 1 y + 3 = v 27 u + 31 v = 85 ........(i) 31 u + 27 v = 89 .........(ii) Adding (i) and (ii) 58 u + 58 v = 174 u + v = 3 .......(iii) Subtracting (ii) from (i) 4 u - 4 v = 4 ⇒ u - v = 1 ........(iv) Adding (iii) and (iv) we get 2 u = 4 ⇒ u = 2 Putting the value of u in (ii) 2 + v = 3 ⇒ v = 1 1 y + 3 = 1 ⇒ y + 3 = 1 ⇒ y = - 2 So, ( x , y ) = 5 2 , - 2

Solve the following simultaneous equations 1 3 x + y + 1 3 x - y = 3 4 ; 1 2 ( 3 x + y ) - 1 2 ( 3 x - y ) = - 1 8

Given, 1 3 x + y + 1 3 x - y = 3 4 ; 1 2 ( 3 x + y ) - 1 2 ( 3 x - y ) = - 1 8 Let 1 3 x + y = u and 1 3 x - y = v u + 2 v = 3 4 and 1 2 u - 1 2 v = - 1 8 So, the equations become 4 u + 4 v = 3 ........(i) 4 u - 4 v = 1 ........(ii) Adding (i) and (ii) = 8 u = 4 ⇒ u = 1 2 Putting the value of u in (i) 1 2 + 2 v = 3 4 ⇒ v = 1 4 1 3 x + y = u and 1 3 x - y = v ⇒ 1 3 x + y = 1 2 3 x + y = 2 .......(iii) Also, 1 3 x - y = 1 4 ⇒ 3 x - y = 4 ........(iv) Adding equation (iii) with (iv) we get, = 6 x = 6 ⇒ x = 1 y = - 1