How can I derive the ratio of concentrations?

I need to derive the equation
$$
 \frac{[\ce{In^-}]}{[\ce{HIn}]}
 = \frac{A\_\mathrm{int} - A\_\mathrm{lo}}{A\_\mathrm{hi} - A\_\mathrm{int}}
\tag1\label{eq1}
$$
from the material balance equation that expresses the conservation of matter principle:
$$c=[\ce{HIn}]+[\ce{In^-}].\tag2\label{eq2}$$


I need to use the above mentioned equation along with the equation
$$
 A^{\lambda} 
 = a\_\mathrm{HIn}^{\lambda} b[\ce{HIn}] 
 + a\_{\ce{In-}}^{\lambda} b[\ce{In^-}]
\tag3\label{eq3}
$$


where $A^{\lambda}$ is the absorbance of an indicator solution whose pH is within the effective pH range.


I will admit, I have no idea how to go about this. I've tried finding this derivation online but I don't even know what exactly to search. Can someone please help.


If I substitute \eqref{eq2} into
$$A\_{\ce{HIn}} = a\_{\ce{HIn}}^{\lambda} b c,$$
then I get
$$A\_{\ce{HIn}} = a\_{\ce{HIn}}^{\lambda} b([\ce{HIn}] + [\ce{In^-}]).$$


What do I do from here? Did I start this correctly?




[![enter image description here](https://i.stack.imgur.com/QHmcPm.png)](https://i.stack.imgur.com/QHmcPm.png)


The signals of the two species add up (linear relationship). If species B absorbs more than species A, call the absorbance of B at a concentration $c\_0$ $A\_\mathrm{hi}$, the absorbance of A at a concentration $c\_0$ $A\_\mathrm{lo}$. A mixture of A and B, with a total concentration of $c\_0$, will have an intermediate absorbance of $A\_\mathrm{int}$. In the example above, 70% B and 30% A has an absorbance that is closer to the absorbance of pure B (70% of the total way from A to B).


$$\frac{A\_\mathrm{int} - A\_\mathrm{lo}}{A\_\mathrm{hi} - A\_\mathrm{lo}}$$ gives you the percentage of B.


$$\frac{A\_\mathrm{hi} - A\_\mathrm{int}}{A\_\mathrm{hi} - A\_\mathrm{lo}}$$ gives you the percentage of A.


Defining "hi" and "lo" is just to avoid negative numbers. You could use the absorbance of A and B straight up for the same type of formula.


To get the ratio, divide the two fractions and cancel the two denominators.


This gives the general idea. For the formal proof, you would calculate the absorbance of pure A, pure B and a mixture using the linear relationship, and then plug that into the expression for the ratio of A and B (rhs of first equation). If all goes well, the ratio of A to B will result after simplifying the longish expressions.


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