1. For all non-zero numbers x and y such that x= 1/y, what is the value of ( x- 1/x)(y+ 1/y)
if x = 1/y , then xy = 1
(x-1/x)(y+1/y)
= (x^1 -1)/x * (y^2 + 1)/y
= (x^1 - 1)(y^1 + 1)/xy
= (1/y^2 - 1)(y^2 + 1)/1
= 1 + 1/y^2 - y^2 - 1
= 1/y^2 - y^2
or
= ( 1 - y^4)/y^2
or more directly
( x- 1/x)(y+ 1/y)
= xy + x/y - y/x - 1/(xy)
= 1 + (1/y)/y - y/(1/y) - 1
= 1/y^2 - y^2
as above
Well, let's start with the equation x = 1/y. To find the value of (x - 1/x)(y + 1/y), we can substitute x with 1/y:
(1/y - 1/(1/y))(y + 1/y)
Now we can simplify it:
(1/y - y)(y + 1/y)
Combining the terms:
((1 - y^2)/y)(y + 1/y)
Finally, simplifying further:
(1 - y^2)(y^2 + 1) / y^2
We now have the value for (x - 1/x)(y + 1/y), which is (1 - y^2)(y^2 + 1) / y^2.
To find the value of (x - 1/x) * (y + 1/y), we need to substitute x = 1/y into the expression first.
Since x = 1/y, we can rewrite the expression as:
(1/y - 1/(1/y)) * (y + 1/y)
Now, let's simplify this expression step by step.
First, let's simplify the numerator of the first term:
1/y - 1/(1/y) = 1/y - y/1 = 1/y - y
Next, let's simplify the denominator of the second term:
y + 1/y = y/(y/1) + 1/y = y^2/y + 1/y = (y^2 + 1)/y
Now, we can rewrite the expression as:
(1/y - y) * (y^2 + 1)/y
To simplify further, we can multiply the fractions:
[(1 - y^2) * (y^2 + 1)] / y^2
Now, let's expand the numerator:
(1 - y^2)*(y^2 + 1) = y^2 + 1 - y^4 - y^2 = -y^4 + 1
Therefore, the value of (x - 1/x) * (y + 1/y) is given by:
(-y^4 + 1)/y^2
To find the value of (x - 1/x)(y + 1/y) where x = 1/y, we need to substitute the value of x in terms of y into the expression.
Given that x = 1/y, we can substitute x in the expression as follows:
(x - 1/x)(y + 1/y) = (1/y - 1/(1/y))(y + 1/y)
Now we have a complex fraction, but we can simplify this.
To combine the fractions, we need a common denominator. The common denominator between y and (1/y) is y. So, multiplying the numerators by y, we get:
(1/y)(y) - (1/(1/y))(y) = 1 - y^2
Now we have (1 - y^2)(y + 1/y)
To further simplify, we can factor out the difference of squares, which is (1 - y)(1 + y):
(1 - y)(1 + y)(y + 1/y)
And that is the simplified expression for (x - 1/x)(y + 1/y) where x = 1/y.