Question

Why do linear equation in two variable has infinte roots?

Answer 1

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Consider the equation ax+by=cax+by=c.

Now we know that there is a number d=lcm(a,b)d=lcm(a,b).

Let ad1=d and bd2=dad1=d and bd2=d. So, ad1−bd2=0ad1−bd2=0. Note that neither d1d1nor d2d2 is 00.

ax+by+ad1−bd2=cax+by+ad1−bd2=c

a(x+d1)+b(y−d2)=ca(x+d1)+b(y−d2)=c

Thus, for every solution of this linear diophantine equation of two variables, there exists another unique solution. Hence, there is an infinite number of solutions to this equation.