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Linear Equations:
A linear equation is any equation of several variables and of course the geometric solution of a linear equation has to be a line. An example of a simple linear equation would be:
Another example:
If you noticed, in the second example we don't have all the variables present. Why? Well it is just because the coefficients in front of some variables are equal to 0. We can rewrite the second example as following:
So in general any linear equation with n unknowns would look as following:
The coefficients a1,2,3,...,n and the constant c are known. Variables x1,2,3,...,n are unknown. So in our second example we have 7 unknowns and each unknown has its own coefficient. The constant c is equal to 257.
So how can you tell which equation is linear and which is not? Basically you cannot have any powers on your unknown variables other than 1, nor can you have two variables multiplied together. Also you cannot have any trigonometric equations (sin, cos, etc). Let me just illustrate few examples of non-linear equations:
None of the above equations are linear. All your unknown variables in a linear equation have to be to the power 1.