Determining rate constants for parallel reactions from experimental data

I have a question regarding the rates of parallel reactions. I have gathered experimental data for the overall reaction (i.e. conversions and selectivities and specific times) but due to the two reactions running parallel to each other for this reaction means I can't use a simple integrated rate plot.


Say for


$$
\ce{A ->[$\large k\_1$] B} \\
\ce{A ->[$\large k\_2$] C}
$$


By substitution and integration, I have found that:


$$
[A](t) = [A](0) \mathrm{exp}(-(k\_1+k\_2)t)
$$


and then the following equations can be derived in terms of the products formed:


$$
\frac{d[B]}{dt} = k\_1 [A](0) \mathrm{exp}(-(k\_1+k\_2)t)
$$


which gives


$$
[B](t) = \frac{k\_1[A](0)}{k\_1+k\_2}(1-\mathrm{exp}(-(k\_1+k\_2)t)
$$


when integrated. Also,


$$
[C](t) = \frac{k\_2[A](0)}{k\_1+k\_2}(1-\mathrm{exp}(-(k\_1+k\_2)t)
$$


is the equation for the other parallel reaction.


My initial thoughts were to plot the graphs (as I know the values for $[C]$, $[B]$, and $[A]$) and form some simultaneous equations, but this is an exponential plot and so i'm not sure how this would work.. or if I could even plot it, as I don't know $k\_1$ and $k\_2$ - that is what I need to determine.




One way you could get $k\_1$ and $k\_2$ is to first find their sum using your equation for $[A](t)$ at a known $t$.


Once you have done that, you can solve for $k\_1$ using your equation for $[B](t)$ at a known $t$, since you now have the denominator of your first factor. You could alternatively find $k\_2$ using $[C](t)$.


Once you have $k\_1$, then $k\_2$ is easily found from the sum. 


 3