Question 330700
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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 *[tex \LARGE \mathbb{R}^2], except linear equations that graph to a vertical line, namely equations of the form *[tex \LARGE x\ =\ \alpha] where *[tex \LARGE \alpha] is any real number, are functions.  The discussion gets a bit more complex for *[tex \LARGE \mathbb{R}^n,\ n\,\geq\,3] 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 <i><b>an</b></i> 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 *[tex \LARGE \mathbb{R}^2] space anyway, one for every real value of *[tex \LARGE \alpha] in *[tex \LARGE x\ =\ \alpha].



John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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