A linear equation has the following form: 

y = mx + b 

where 

m is the slope
b is the y-intercept. 

You can also perform a vertical line test. If the line touches your graphed function in more than one spot, it is not a 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)

The way how you differentiate a linear and a non linear function is as under-

In a linear equation the variables appear in first degree only and terms containing product of variables are absent.

e.g. y= 2x+3,

  y= -3x+4,

 3 y=2x-4  etc.

But in case of non linear equations at least one variable is not of the first degree or the equation contain product of variables.

e.g. y=x^2+2,

or, y^2=2x-4,

or,   y=2x+3xy-4,

or,   xy=1  etc.

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An equation is linear if its graph forms a straight line.  This will happen when the highest power of x is "1".

Here are a few examples of linear equations:

3x + 2y = 8

y = 2x + 3

y - 2 = 3(x - 1)

(note: all variables are raised to the first degree)

Here are some examples of non-linear equations:

y = x^2 (note: x is raised to the second power)

x^2 + y^2 = 4 (note: both x & y are raised to the second power)

An equation is linear if its graph forms a straight line.  This will happen when the highest power of x is "1".

Graphically, if the equation gives you a straight line thenit is a linear equation. Else if it gives you a circle, or parabola or any other conic for that matter it is a quadratic or nonlinear equation.

ALTERNATIVE:

If the highest power of x in the equation(in x) is 1 then it is a linear equation else if the power of x is greater than 1 then it is nonlinear.(ie IF AND ONLY IF THE COEFFICIENT OF x HAVING THE HIGHEST POWER  IS NONZERO!!!)

Here's one suggestion/method I haven't seen mentioned yet: it's called the vertical line test. This method works well for lines and curves that are already graphed. The vertical line test states that if you draw a vertical line through the line or curve in question and the vertical line intersects the line or curve in only one place, then the line or curve is a function.

One thing I did not see mentioned in any of the above posts is this: a line that is vertical IS NOT a function. So all lines of the form x = a, where a is some number, are not functions. These lines are straight, but do not have a unique x value for each value of y, therefore, not a function.

Also, understand that a line or curve can be a function without being a linear function. For example, ax^2 +by + c = 0 is a function (quadratic function) but not a linear function.

Hello Schnoop,

Determining whether your equation is a linear or non-linear function can be achieved many ways. First off, are you looking at a graph of the function, or is it an equation you are looking at? If it is a graph, then the simplest answer is "Is the graph a straight line?" You see, linear means that all your variables are only to the first power, and when variables are only to the first power, their graphs will always make a straight line. Look at the word LINEar, and notice that the first part says "LINE". That's an easy way to remember linear=line when graphed.

In equation form, is the x, y, or whatever variables you are using to the power of 1, or first power? If you are looking at the equation and the variables are only to the first power, or in other mathematical words to the first degree, then your function is linear.  In short, find all the variables and makes sure that they don't have any exponents attached to them other than a 1.

I hope this helps you, good luck :)

The easiest way I have for knowing the difference between linear and nonlinear is the exponent value on the variable x.

It is important to understand the root word in linear. It is LINE. A straight line, no curves.

For example:

y = 2x - 3 This is linear because the exponent on x is one. Thus your slope is standard rise over run, like a stair step and simply goes up or down.

Y = x^2 + x + 4 is nonlinear. When graphed it becomes a parabola, which looks like a hill on your graph. This is because the exponent on the variable of x is more than one.

This pattern continues on for all equations. Hope this helps to put it into easy to understand terms. :)