Hopefully, I am in the right place. It's been a while since I did any math, so this might be embarrasing...
I have an equation, where I am trying to solve for X.
$Reach = \frac{X * AQH * CUME}{X * AQH + CUME - AQH}$
How do I solve for X?
Hopefully, I am in the right place. It's been a while since I did any math, so this might be embarrasing...
I have an equation, where I am trying to solve for X.
$Reach = \frac{X * AQH * CUME}{X * AQH + CUME - AQH}$
How do I solve for X?
Multiply both sides by the denominator:
$Reach(X \cdot AQH + CUME - AQH)=X \cdot AQH \cdot CUME$
Group the X's:
$X(Reach \cdot AQH-AQH \cdot CUME)=-Reach(CUME-AQH)$
Divide by what's next to X:
$X=\frac{Reach(AQH-CUME)}{AQH(Reach-CUME)}$
$x\cdot Reach\cdot AQH + Reach\cdot(CUME-AQH)=x\cdot AQH\cdot CUME$ $x(Reach\cdot AQH -AQH\cdot CUME)=-Reach\cdot(CUME-AQH)$ $x=\frac{-Reach\cdot(CUME-AQH)}{Reach\cdot AQH -AQH\cdot CUME}=\frac{Reach\cdot(AQH-CUME)}{AQH(Reach-CUME)}$
Say we have a linear fractional equation of the form
$y=\frac{ax+b}{cx+d}.$
We can multiply out:
$y(cx+d) = ax+b$
Subtract the right side and recombine:
$(cy-a)x+(dy-b)=0$
and then subtract and divide for $x$:
$x=\frac{-dy+b}{cy-a}.$
Your particular equation has $y=\mathrm{reach}, a=\mathrm{AQH}\cdot\mathrm{CUME}, b=0, c=\mathrm{AQH}, d=\mathrm{CUME}-\mathrm{AQH}.$
More generally (you might not have the background for this), we can define an action of $\rm SL$ by
$\begin{pmatrix}a & b \\ c & d \end{pmatrix}x=\frac{ax+b}{cx+d}.$
Lo and behold, it is well-defined, i.e. $A(Bx)=(AB)x$ and so $y=Ax$ implies $x=A^{-1}y$ when $A$ is an invertible matrix. Note that we speak of $\rm SL$ instead of $\rm GL$ because $A$ and $cA$ define the same transformation for any $c\ne0$, hence the quotient.