Transformations, Algebra Trouble

[Linday12]

Homework Statement

If f(x)=\frac{2x+1}{x+2}, the equation for y=f^-1(x) is?
So I switch x, x=\frac{2y+1}{y+2}

The Attempt at a Solution

I've tried many ways, but I must be going wrong somewhere, here's what I think to be my nearest:
x(y+2)=2y+1
x(y)+2x=2y+1
x(y)+2x-2x=2y+1-2x
x(y)=2y+1-2x
x(y)-2y=1-2x
And that's as far as I get, it seems I'm stuck in an endless x loop or it simplifies too much to get the correct answer, so I must be missing a vital step or rule.
Sorry, it has been a while since I've done this, and I'm not quite sure if I am even attempting it right. Any tip would be very helpful, I think the answer is:
y=\frac{1-2x}{x-2}

Thank you.

 

[Cyosis]

The answer is correct.

x(y)-2y=1-2x

y(x-2)=1-2x

 

[Linday12]

Interesting. Let me see if I understand.

x(y)-2y=1-2x <----Am I to factor(not sure if that's the proper word) the left side to get that final answer then, similar to what you showed?

And then I get y(x-2) after the factor
Which then is y(x-2)=1-2x
And then I can get my final answer of:
y=\frac{1-2x}{x-2}

So that is all I was missing? Then it makes sense now. Thank you very much!

 

[Cyosis]

Yep just factor and then divide by x-2 on both sides.