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A linear equation is of the general form y = mx + b. This is a function if f(x) gives a unique value.
If y = f(x) = k, where k is a constant, for any value of x, y = f(x) has the same value. As a result y = k is a linear equation but not a function.
A linear equation does not represent a function when there is more than one value of x that gives the same value of y.
A linear equation (a function of the form y = mx + b) is always a function. A function has the property that each input (this is x in the equation) has exactly one output (this is y in the equation).
Therefore, in order for a linear equation to not be a function, there would have to be a case when the same x produces two different y's. This is never the case with a linear equation.
Suppose, for contradiction, that that the linear equation y = mx + b, is not a function. Then there exists two distinct values u and v such that u = mx + b and v = mx + b. But u=v, a contradiction, therefore it is a function.
Functions and linear equation have a common thing among them, that they both deal with 'x' and 'y' coordinates and points on a graph. Still there are differences between them in limitations, form and purpose. Our purpose here is to know “when is a linear equation not a function”. Often, functions are found to provide you with the value of either x or y, but the linear equations ask to solve for both 'x' and 'y'.
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