Why can we use inspection for solving equation with multiple unknowns? In our algebra class, our teacher often does the following: 
$a + b\sqrt{2} = 5 + 3\sqrt{2} \implies \;\text{(by inspection)}\; a=5, b = 3
$
I asked her why we can make this statement. She was unable to provide a satisfactory answer. So I tried proving it myself. 
$a + b\sqrt{2} = x + y\sqrt{2}$. We are required to prove that $a = x$, and $b = y$. Manipulating the equation, we get $\sqrt{2}(b - y) = x - a$, or $\sqrt{2} = \frac{x-a}{b-y}$. Expanding this, we get $\sqrt{2} = \frac{x}{b-y} + \frac{a}{b-y}$. I tried various other transformations, but nothing seemed to yield a result. 

HINT:
Assuming $a,b,x,y$ are rationals $\sqrt2(b-y)=x-a$ rational which is only possible 
if $b-y=0$ 
As for $b-y\ne0,\sqrt 2=\frac{x-a}{y-b}$ which is rational 
