inverse of $y=\frac{5x-3}{2x+1}$

Question

I have solved it as follows:

$\displaystyle x=\frac{5y-3}{2y+1}$

$5y-3=2xy+x$

$5y-2xy=3+x$

$y(5-2x)=3+x$

$\displaystyle y=\frac{3+x}{5-2x}.$

$\displaystyle {f^{-1}}(x)=\frac{3+x}{5-2x}$

That is my answer. But On the screen, there appeared another answer due to a slightly different method of isolating the second stroke:

$2xy+x=5y-3$

$2xy-5y=-x-3$

$y(2x-5)=-3-x$

$\displaystyle y=\frac{-3-x}{2x-5}$

$\displaystyle {f^{-1}}(x)=\frac{-3-x}{2x-5}$.

As you can see there are two answers now. So which is the real answer?

Answer 1

$\displaystyle y = \frac{5x-3}{2x+1}$

$\displaystyle (2x+1)y=5x-3$

$\displaystyle 2xy+y=5x-3$

$\displaystyle x(2y-5)=-y-3$

$\displaystyle x(5-2y)=y+3$

$\displaystyle \boxed{x=\frac{y+3}{5-2y}}$

Hope this helped. Have a nice day :D

Both answers are correct. Multiply by $\displaystyle \frac{-1}{-1}$ to get from one answer to the other.

Answer 2

This method is not applicable always , but sometimes , it helps very much when you are having confusion whether two quantities are equal or not

Just set $x=1$ or any natural value in both expression, for simpler calculation , and see whether both are giving same results or not

In your case :

putting $x=1$ in Expression $1st$=$${{3+x}\over {5-2x}}={{3+1}\over {5-2(1)}}={4\over 3}$$

similarly putting $x=1$ in expression $2nd$=

$${{-3-x}\over {2x-5}}={{-3-1}\over {2(1)-5}}={-4\over -3}={4\over 3}$$

Both expressions are resulting same at $x=1$ , so both should be equivalent

But as I said, it doesn't works always , but most of times it is useful in comparing equality of two quantities