How do I determine if an expression is linear?

Question

I want to determine if a multi-variable expression is linear. My current test is to take the second derivative of the expression to see if it is 0.

Given the expression x-y = 0:

d^2(x-y)/(dxdy) = 0 (linear)

However, if I rearrange to x/y=1:

d^2(x/y)/(dxdy) = -1/y^2 (non-linear)

x-y = 0 is equivalent to x/y=1 however, depending on how I have arranged the equation one appears linear and one does not. Is there a method independent of expression arrangement which determines linearity?

Answer 1

The question is flawed from the beginning, as

$$x-y=0$$ is not an expression but an equation.

You may test the linearity of the expression $$x-y$$ but then the ratio $$\frac xy$$ is just irrelevant.