Reverse Process

Level pending

If $f$ $(\dfrac{x-3}{5-x})$ $= 3x - 2$

Find $f (x)$ .

The answer can be written in the form of $\dfrac{Ax + B}{Cx + D}$ , such that $A, B, C, D$ are all integers. Find $A + B + C + D$ .

*Kindly indicate your solution if possible.

1 solution

Emmanuel John Baliwag
Apr 15, 2015

Let ( $\dfrac{x-3}{5-x}) = n$

Solve for $x$ in terms of $n$

( $\dfrac{x-3}{5-x}) = n$

$x - 3 = 5n - nx)$

$5n + 3 = x +nx$

$5n + 3 = x(n +1)$

$\dfrac{5n+3}{n+1} = x$

Now substitute this to $3x - 2$

$3(\dfrac{5n+3}{n+1}) - 2$

$\dfrac{15n+9}{n+1} - 2$

$\dfrac{15n+9}{n+1} - \dfrac{2n +2}{n+1}$

$\dfrac{15n+9 - 2n - 2}{n+1}$

$\dfrac{13n - 7}{n+1}$

The answer asks $A + B + C + D$ or $13 + 7 + 1 + 1$ which is equal to $\boxed{22}$