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If it can be written in the form y = mx + c where m and c are constants [or, equivalently, ax + by = k where a, b and k are constants] then y is a linear function of x.
The word linear means in a straight line. If the graph is a line, it is linear. Also, linear equations are of the first order; they contain a variable but not a square (or higher power) of a variable. If the equation contains x2 it is not linear.
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.
A linear function would be represented by a straight line graph, so if it's not a straight line, it's nonlinear
The "vertical line test" will tell you if it is a function or not. The graph is not a function if it is possible to draw a vertical line through two points.