How do I determine if this equation is a linear function or a nonlinear function?

The variable x must be either degree zero or degree 1 AND the variable y must be 1st degree in order to be a linear function.

Examples:

y = 2x - 3 (both x and y are 1st degree)

4x + 5y = 20 (both x and y are first degree)

2x - 4y = 7 + 3x (all variables are 1st degree)

y = -1  (x is degree zero and y is 1st degree; this makes a horizontal line which is a function of x)

If variable x is 1st degree but the variable y has a degree of zero, it will be a linear relation but not a function of x.

Example:

x = 4  (the graph is a vertical line and is not a function of x)

If variable y is 1st degree but the variable x has a degree other than 0 or 1, it will be a non-linear function of x.

Examples: 

y = x^2 + 25 (x is not first degree)

y = 5x + 2 - x^3 (x is 3rd degree)

y = 1/x or y = x^(-1)  (x is to the power of -1)

y = sqrt(x) or y = x^(1/2) (x is to the 1/2 power; the graph is 1/2 a sideways parabola)

y = 2^x  (x is the exponent instead of the base, so the graph is exponential and not linear)

If variable y is not 1st degree, the relation will not be a function of x.

Example: 

x^2 + y^2 = 4 (neither x nor y is 1st degree; the graph is a circle with a radius of 2)

x = y^2 (y is not 1st degree; this is a sideways parabola)