Correspondence between variables in the equation $Ax+By = Cx+Dy$

Question

If $Ax+By = Cx+Dy$ is it true to say that there always must be $A=C, B=D$? all variables are real numbers

Answer 1

(I'm assuming all variables are supposed to be real numbers.)

Yes, if your equation is supposed to hold for all real numbers $x$ and $y$, then $A=C$ and $B=D$. This line of reasoning is called comparison of coefficients or equating coefficients.

If your equation only holds for some particular real numbers $x$ and $y$ then this does not have to be true.