How do I connect these two parts of solution?
Please double check.

I find your equations very confusing as they have Many factors on each side of the equation sign. 
I found :
by applying
\$\Delta V_{b}=\Delta R_{s}\cdot i_{b}\$

\$Ad\cdot\Delta V_{b}=\Delta V_{c}\$

replacing with known values.

\$\frac{Rc}{r_{e}}\cdot\Delta R_{s}\cdot\frac{I}{2\beta}=\Delta I\cdot R_{c}\$

you get

\$
\Delta R_{s}\cdot\frac{1}{r_{e}}\cdot\frac{I}{2\beta}=\underline{\underline{\Delta I}}
\$

As a note I might be a factor 2 off.. But it seems alright.
This is also a low entropy solution, where you see what other factors affects \$\Delta I\$
Hope it helps .