Why can we use inspection for solving equation with multiple unknowns?

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.