Trying to solve this system of simultaneous equations. A solution with work shown would be appreciated.

You may rewrite the system of equations as

$ae^{-bc}e^{21.55b}+d=52.45 $
$ae^{-bc} e^{44.41b}+d=107.594$
$ae^{-bc}e^{53.627b}+d=192.024 $
$ae^{-bc}e^{57.056}+d=271.882$.

Let $ae^{-bc}=x$. Then 

(1)   $xe^{21.55b}+d=52.45 $
(2)   $xe^{44.41b}+d=107.594$
(3)   $x^{53.627b}+d=192.024 $
(4)   $xe^{57.056}+d=271.882$.

(2)-(1) gives 
\[ (5)              x(e^{44.41b}-e^{21.55b})=55.144.\]
(4)-(3) gives 
\[ (6)         x(e^{57.056b}-e^{53.627b})=79.858.\]
Dividing (6) by (5) we get 
\[\frac{e^{57.056b}-e^{53.627b}}{e^{44.41b}-e^{21.55b}}=\frac{79.858}{55.144}.\]
This equation can not be solved analytically, but it is easy tobe solved numerically (you can simply graph this function to find the root). We get
\[b=0.113.\]
You can substitute $b$ in (5) to get $x$, and then use (1) to compute $d$. So you can  only computer $b, d$, and $x$. But the parameters $a$ and $c$ can not be computed from 
\[x=ae^{-bc}.\]

I hope this helps.