Simultaneous equations?

Question

y=(x-1/x)^x
y=x
What value is true for x and y (can you express it exactly?)?

Seems I've caused a little bit of confusion. So the line y=x is known right? For the former one I mean treat x minus its reciprocal as a single number. We shall call this single number a. Now raise a to the power of x and I'm certain these lines cross. I've plotted them myself but can't figure out where they intersect.

Answer 1

1.92477079652487 approximates the only Real root.

There can be only one real root, too. If x>=2 then (x-1/x)^x lies avbove the graph y=x. If x<1 then there are discontinuity breaks (for instance x=.7 is undefined). By graphing the region from 1
There will be, however, lots more complex solutions if x is not strictly a real number. Complex points where (z-1/z)^z is analytic in z will yield solutions.

Also, the solution is transcendental so there will be NO algebraic representation for the solution (an algebraic soltion is one that solves a polynomial; e.g. x=4, x =(2-sqrt(5))^(1/2) are algebraic representations).

Answer 2

we only need to consider a bounded region where -2 < x < 2 because at those values, y is extreme enough and trending consistently that there is no intersection. We know that the first equation has no solution at x=0 because of divide by 0.
The answer of 1/9 is wrong, because when x = 1/9, y = -1.274757..... (1 - 1/(1/9) = 1-9 = -8. the 9th root of -8 can't be positive (of that i am positive).

I believe that there is no value for y where the two equations intersect. The error in using logarithms occurs because you cannot take the log of a negative number, no?

Answer 3

When drawn the values of x and y are 1/9, 1/9

Answer 4

I reckon x and y = -1/9

So:
y=(x-1/x)^x and y=x
so
x=(x-1/x)^x
so
x=xlog((x-1)/x)
1=log((x-1)/x)
so
10^1=(x-1)/x
10x=x-1
10x-x=-1
9x=-1
x=-1/9
y=x
y=-1/9

I think this is right!

Ashley
EDIT:
This doesn't to work when put back into the equation, but I dont know where I went wrong.

Also, using my graphic calculator: there is no answer, they dont cross!

Mail me if you find the right answer will you?

Answer 5

type them both into your graphing calculator, where they intersect is the answer.

Answer 6

y≈1.9247708716703
x≈1.9247708778947
and you can check it your self

Answer 7

y=(x-1/x)^x and y=x

x = [(x-1) / x]^x
x = (x-1)^x / (x^x)
x^(x+1) = (x-1)^x
x^[(x+1)/x] = x-1
[(x+1)/x]lgx = lg(x-1)
(x+1)/x = lg(x-1)/lgx

OR
x = (x - (1/x))^x
x = [(x^x)(x - (1/x))^x] / (x^x)
x = [[x(x - (1/x))]^x ] / (x^x)
x = [[x^2 - 1]^x] / (x^x)
x = [(x^2 - 1)/x]^x
x = [(x-1)(x+1)/x]^x

boy... I'm stuck!!!