Can anyone help me solve this problem?

Question

Simplify the expression

x + 1/x - a - 1/a
---------------------
.........x-a

the answer is

ax-1 / ax

I just need to see all the steps

Answer 1

first i multiplied the top and bottom by ax

ax^2 + a - a^2(x) - x / [(-ax)(a-x)]

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

Answer 2

Very interesting problem. First, start by getting a common denominator on the top: ((ax^2+a-a^2x-x)/xa)/x-a). than group the top together ax(x-a)-1(x-a), which than simplifies to (ax-1)(x-a). Now put it all together and u have (((ax-1)(x-a))/(ax))/(x-a). The (x-a) cancel on top and bottom, leaving you with (ax-1)/(ax)

Answer 3

simplify numerator

x + 1/x -a - 1/a = (ax^2 +a -a^2x - x)/ax = (x-a)(ax-1)/ax
cancel the (x-a) factor with the orgininal denominator to get
(ax-1)/ax = 1 - 1/ax

Answer 4

x/x = 1 a/a=1 and ax/ax = 1
multiplying by 1 does not change a value:

top:
x + 1/x - a - 1/a
x(ax/ax) + (1/x)(a/a) - a(ax/ax) - (1/a)(x/x)
axx/ax + a/ax - aax/ax - x/ax
(easier to write xx than x^2)

Now, everything is over the same denominator (ax) which I send to the bottom, leaving me with this on top:
axx + a - aax - x
and on the bottom: ax(x-a)

Top:
axx + a - aax - x
axx - x + a - aax (rearranging)
(axx - x) + (a - aax) (just added brackets without changing anything)
x(ax - 1) + a(1 - ax)

(1 - ax) is the same as -(ax - 1); check it out.
x(ax-1) - a(ax-1)
(x-a)(ax-1)

Back to the entire fraction:

(x-a)(ax-1) / ax(x-a)

(ax -1) / ax
QFD

Answer 5

Multiplying numerator and denominator by ax gives:
(ax^2 + a - a^2x - x) / ax(x - a)
= ( ax(x - a) + (a - x) ) / ax(x - a)
= (x - a)(ax - 1) / ax(x - a)
= (ax - 1) / ax.

Answer 6

this is not very hard but you must remember the formulas.
I thing you must multiply the both parts (up and down) with x and then with a
but I'm not sure