$ax+by=cx+dy$, where $x$ and $y$ are variables. Does $a=c$, $b=d$?

Question

If I have the equation $ax+by=cx+dy$, where $x$ and $y$ are variables and $a,b,c,d$ are coeficients. And also this equation holds for all $x$ and $y$. Can we conclude from this equation that $a=c$ and $b=d$? If we can, is there some theorem about that or proof?

Answer 1

If it holds for all values of $x,y$ then it holds for $x =1; y = 0$ and for $x=0; y=1$.

So $a*1 + b*0 = c*1 + d*0$ and $a*0 + b*1 = c*0 + d*1$.

Answer 2

If this equation needs to be hold for arbitrary $x$ and $y$, then it is a ‘YES’; otherwise it is a ‘NO’

Answer 3

write your equation in the form $$x(a-c)=y(d-b)$$ if this is fulfilled for all real $x,y$ then it must be $$a=c$$ and $$b=d$$