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To be linear, there should only be constants, and variables with constant coefficients. No powers of variables, or numbers raised to the power of a variable, or any other nonlinear function such as log, ln, sin, cos, tan, cosh, etc.
A linear function would be represented by a straight line graph, so if it's not a straight line, it's nonlinear
You can easily tell by substituting 0 for a.
If the two equations are linear transformations of one another they have the same solution.
An equation is a statement that two things are equal. A function is a rule or process that gives you a value if you give it something in its domain (the set of things on which it is defined) as an argument. Functions on numbers that are defined by a rule can usually be expressed by an equation. A linear function is one that can be defined by a linear equation.