Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am having a bit of trouble solving for x when trying to find $f^{-1}$. I have $$y=\frac{x+5}{x-4}$$ How can I get x on one side? I tried multiplying both sides by the denominator but I am still left with an $x$ on both sides...

share|improve this question
    
That was a good start! After that move the terms containing an $x$ to the other side. Remember to swap signs when appropriate. –  Jyrki Lahtonen Sep 3 '12 at 6:36

4 Answers 4

up vote 3 down vote accepted

You just need to expand out the brackets and re-factor:

$$ \begin{array}{rcl} y & = & \frac{x+5}{x-4}\\ y(x-4) & = & x+5\\ yx-4y&=&x+5\\ yx-x &=&4y+5\\ x(y-1)&=&4y+5\\ x&=&\frac{4y+5}{y-1} \end{array} $$

share|improve this answer

Note that $x+5 = 1\cdot(x-4) + 9$. Just as with numbers, this means that $x-4$ divides $x+5$ once with a remainder of 9. So you can write $(x+5)/(x-4)$ as $1+\frac{9}{x-4}$. It should be easy to get $x$ on one side now.

share|improve this answer
1  
For doing $x+5=1.(x-4)+9$, assuming $x\neq 4$ is essential. :-) –  Babak S. Sep 3 '12 at 6:38
    
Just as with numbers, one needs to be careful when dividing by zero. Simply by paying careful attention to the statement of the problem we can see that $x=4$ could be problematic. –  shoda Sep 3 '12 at 17:18
    
Usually when we are working with rational functions; we delete the roots of its denominator of our domain. So, what you did here is someway legal. –  Babak S. Sep 3 '12 at 17:35

First multiply out to get rid of the fraction: $y(x-4)=x+5$, or $yx-4y=x+5$. Now collect the $x$ terms on one side and everything else on the other: $yx-x=5+4y$. Factor the lefthand side: $(y-1)x=5+4y$. Now just divide both sides by $y-1$, and you’ll have $x$ in terms of $y$.

share|improve this answer

Multiply with the denominator:

$$y(x-4)=x+5$$ $$xy -4y=x+5$$ Take all x and bring them to the left side, all other terms to the right side. $$xy -x = 5+4y$$ Single out x: $$x(y-1)=5+4y$$ Divide by the factor: $$x=\frac{5+4y}{y-1}$$

Note: $x\ne4$, because of the first equation.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.