prove that slope is unique (using geometric definition).please give me
a detailed proof by using proof by contradiction.
thank you

Hi,

The slope of a line joining point A(x1,y1) and point B(x2,y2) is defined as:

slope = (y2-y1)/(x2-x1)

Geometrically this is equal to: rise/run 
i.e. how much the graph has increased over the distance it took to increase.

Proof by contradiction works by assuming the contrary of what you want
to prove, then prove that such an assumption is false.

Proof:

Let us assume that there is another slope value for a given graph. If
this was true, then either the numerator or the denominator (or both)
would have to be different.

Let us assume that the numerator is different, this would indicate
that a function can take two different values for the same parameter.
This contradicts the definition of a function. Therefore the premise
that the numerator is different is false.

Let us assume that the denominator is different, then we wouldn't be
calculating the slope over the same interval, therefore we wouldn't be
analyzing the same graph. Thus this premise is also false.

Therefore our original assumption must be false. There cannot be a
different slope value for a given graph.

Conclusion:
Slope is unique for a given graph.


I hope this is clear enough. If you require any clarifications, please
do not hesitate to ask.

Thanks.
endo