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Show that the slope of the segment with the endpoints ( 1/x ,1 )and ( 1/y, y/x ) does not depend on x. Provide your complete solutions and proofs in your paper homework and respond to questions or statements online.

Respuesta :

jcherry99

Answer:

Step-by-step explanation:

Givens

y2 = y/x

y1 = 1

x2 = 1/y

x1 = 1/x

Formula

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

Solution

m = (y/x - 1) / (1/y - 1/x)

m = ( (y - x)/x ) / ((x - y)/xy)

m = (-(x - y) / x) / ( (x - y) / xy)

Invert the denominator and multiply

m = ( - (x - y) / x ) * x*y / (x - y)      Notice x - y cancels.

m = ( - 1 / x ) * (x*y)                        Notice the xs will cancel.

m = ( - 1 / y)                                   There are no xs anywhere. The slope is - 1/y

facundo3141592

Answer: Hello there!

If you have the set of points (x1, y1) and (x2, y2), the slope of the segment whit this endpoints is:

[tex]a = \frac{y2 - y1}{x2 - x1}[/tex]

Now, we have the endpoints (1/x, 1) and (1/y, y/x), the slope of this segment is:

[tex]a = \frac{y/x - 1}{1/y - 1/x}[/tex]

Let's simplify it:

[tex]\frac{y/x - 1}{1/y - 1/x} = \frac{(y-x)/x}{(x - y)/x*y} = \frac{\frac{y-x}{x} }{\frac{-(y-x)}{y*x} }= y*\frac{\frac{y-x}{x} }{\frac{-(y-x)}{x} }  = y*(-1) = -y[/tex]

Then the slope of the segment is a = -y, and it does not depend of x