Solve the following simultaneous equations:

103x + 51y = 617, 97x + 49y = 583

Given,

Equations :

103x + 51y = 617     …….(1),

97x + 49y = 583     …….(2)

Adding equations (1) and (2),

⇒ 103x + 51y + (97x + 49y) = 617 + 583

⇒ 103x + 51y + 97x + 49y = 617 + 583

⇒ 200x + 100y = 1200

⇒ 100(2x + y) = 1200

⇒ 2x + y = $\dfrac{1200}{100}$

⇒ 2x + y = 12     ……..(3)

Subtracting equation (2) from (1),

⇒ 103x + 51y - (97x + 49y) = 617 - 583

⇒ 103x + 51y - 97x - 49y = 34

⇒ 6x + 2y = 34

⇒ 2(3x + y) = 34

⇒ 3x + y = $\dfrac{34}{2}$

⇒ 3x + y = 17     ……..(4)

Subtracting equation 4 from 3,

⇒ 2x + y - (3x + y) = 12 - 17

⇒ 2x + y - 3x - y = -5

⇒ -x = -5

⇒ x = 5.

Substituting value of x in equation 1,

⇒ 103x + 51y = 617

⇒ 103(5) + 51y = 617

⇒ 515 + 51y = 617

⇒ 51y = 617 - 515

⇒ 51y = 102

⇒ y = $\dfrac{102}{51}$ = 2.

Hence, x = 5, y = 2.