Solve the following simultaneous linear equations:

97x + 53y = 177

53x + 97y = 573

Given,

97x + 53y = 177 …….(i)

53x + 97y = 573 …….(ii)

Multiplying eq. (i) by 53 and (ii) by 97,

5141x + 2809y = 9381 …….(iii)

5141x + 9409y = 55581 ……(iv)

Subtracting eq. (iii) from (iv) we get,

⇒ 5141x + 9409y - (5141x + 2809y) = 55581 - 9381

⇒ 6600y = 46200

⇒ y = $\dfrac{46200}{6600}$ = 7.

Substituting value of y in (ii) we get,

⇒ 53x + 97(7) = 573

⇒ 53x + 679 = 573

⇒ 53x = 573 - 679

⇒ 53x = -106

⇒ x = -2.

Hence, x = -2 and y = 7.