suggest a simple and short method to solve the following equations

a1*x + b1 *y + c1*xy = d1

a2*x + b2 *y +  c2*xy = d2

plz dont suggest the conventional way of getting value of one of the variables and putting in second equation

Yeah, there is a way if u like it. you can apply cramer's rule used widely in determinants but yeah with some modifications. You can refer any maths competition book for it.