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Question 330700: What similarities and differences do you see between functions and linear equations? Are all linear equations functions? Is there an instance in which a linear equation is not a function?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
A linear equation, in most cases is a function. There are many functions, however, that are not linear. A linear equation relates a dependent variable to one or more independent variables such that a set of ordered pairs defining a straight line is defined. A function, in general, relates a dependent variable to one or more independent variables such that a relation where a given value for the input variable (or set of values) determines a unique value for the dependent variable. Linear equations fit the function definition except in certain special cases.
All linear equations in  , except linear equations that graph to a vertical line, namely equations of the form  where  is any real number, are functions. The discussion gets a bit more complex for  but the idea is the same. If you can define a line that violates the idea of a single value of the function for a single value of the input variable (or set of values for the input variables), then you have a case where you do not have a function.
So, no, there is not an instance in which a linear equation is not a function, there are an infinity of instances in which a linear equation is not a function, namely, in space anyway, one for every real value of in .
John
My calculator said it, I believe it, that settles it
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