How to calculate what temperature change in entropy for the reaction equal to the change in entropy for the surroundings given Enthalpy and entropy
You're looking for the equation of Gibbs free energy:
$$\Delta G^\circ =\Delta H^\circ - T\Delta S^\circ$$
Per Wikipedia's Gibbs free energy page:

The equation can be also seen from the perspective of the system taken
together with its surroundings (the rest of the universe). First
assume that the given reaction at constant temperature and pressure is
the only one that is occurring. Then the entropy released or absorbed
by the system equals the entropy that the environment must absorb or
release, respectively. The reaction will only be allowed if the total
entropy change of the universe is zero or positive. This is reflected
in a negative ΔG, and the reaction is called exergonic.

So effectively you're trying to deduce the $T$ that produces $\Delta G = 0$ which is the &quot;tipping point&quot; (then the reaction will occur for $\Delta G &lt; 0$)
So you solve for $T$ in:
$0 = \Delta H - T \Delta S$
Thereafter as $T \rightarrow \infty$ you see that $\Delta G$ keeps decreasing, making the reaction more favorable.
Thus $S_\textrm{system} = S_\textrm{surroundings}$ is a fancy prerequisite for reaction occurrence, which we commonly associate with $\Delta G \le 0$.