Question

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.

Answer 1

Answer: Hello there!

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

`a = (y2 - y1)/(x2 - x1)`

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

`a = (y/x - 1)/(1/y - 1/x)`

Let's simplify it:

`(y/x - 1)/(1/y - 1/x) = ((y-x)/x)/((x - y)/x*y) = ((y-x)/(x) )/((-(y-x))/(y*x) )= y*((y-x)/(x) )/((-(y-x))/(x) )  = y*(-1) = -y`

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

Answer 2

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