Radioactive decay problem solution

 The other day, I was tutoring a student taking Math 111, and some radioactive decay problems came up, as the student was in the logarithm and exponential function part of the course (towards the end). At the time, the solution I outlined on paper didn't come out right, so I redid the solution on the back of the paper the next day, and arrived at the proper solution (checked by using Microsoft Math Solver on my iPhone). In order to get the solution in a form that the student could decipher, I used the rendering techniques I outlined in an earlier post (using Hostmath). Here's the result, which can be displayed in any browser:

\[32=A_{0}e^{-18k} \] \[10=A_{0}e^{-65k} \] \[A_{0}=\frac{32}{e^{-18k}} \] \[10=\frac{32e^{-65k}}{e^{-18k}}=32e^{-65k+18k}=32e^{-47k} \] \[\frac{10}{32}=e^{-47k}\Rightarrow ln{\frac{10}{32}}=-47k\Rightarrow \frac{ln\frac{10}{32}}{-47}=k \] \[k=0.0247 \] \[32=A_{0}e^{-18(0.0247)}\Rightarrow 32=A_{0}e^{-0.4446} \] \[A_{0}=\frac{32}{e^{-0.4446}}=\frac{32}{0.6411}=49.91 \]

The problem went something like this: A radioactive substance had 32 g left after 18 days and 10 g after 65 days. How much did we start out with, what is the half-life of the substance, and how much will be left after 100 days? I was just answering the first part of the question. The solution involves solving a system of two equations in two unknowns, and I used the substitution method for the above solution.

It didn't take a long time to do the Hostmath stuff, but it had to be done line by line. Still looking for an even better solution. If you know of one, please put it in the comments. Thanks!