How do you simplify: a-x/ax(x-a)

Question

I know the answer is -1/ax, but I'm not quite clear on the rules for simplifying it.

Answer 1

Be careful how you write something like this. 

 

If you have two terms here, a, and another you're subtracting from it, x/ax(x - a), you'll get something different than if you have both a and x in the numerator as you would in (a - x)/ax(x - a).

 

It's hard to write fractions on this website. 

 

But there's a difference between writing 

 

a - x/ax(x - a)         and 

 

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

 

If you want to simplify the first one you would reduce that second term by noting that it has an x on the top and on the bottom. Whenever a fraction has the same factor in the numerator and denominator, you can reduce it by dividing that factor out of each.

 

So x/ax(x - a) becomes 1/a(x - a).

 

Subtract this from a:

 

a - 1/a(x - a)                          combine the terms into one fraction

 

[a(a(x -a)) - 1]/a(x - a)             simplify the numerator              

 

[a²x - a³ - 1]/a(x - a)                and I'm going to stop there, because this probably isn't what you meant...

 

 

 

I'm going to assume now that we're dealing with

 

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

 

Note that you have (a - x) on the top. You can rewrite this by factoring out -1

 

(a - x) = -1(x - a)

 

Then 

 

(a - x)/ax(x - a)        becomes

 

-1(x - a)/ax(x - a)     divide out the common factor, x - a

 

-1/ax                        as long as x doesn't equal a

 

 

Answer 2

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

 

Note that (a-x) = -(x-a).  Let's substitute that into the expression:

 

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

 

The (x-a) terms cancel, leaving you with:

 

-1/ax