Prove that there exists a canonical isomorphism of A-algebras [closed]

Just to write an answer I'll rewrite my comment here.

The fact that we are quotienting out by $\langle t_iX_i-1\mid i\in I\rangle$ implies that in the quotient we have $t_iX_i-1=0$ for all $i$, which is equivalent to $t_iX_i=1$. In other words, $X_i$ is the multiplicative inverse of $t_i$. This forces $t_i^{-1}\in S^{-1}$ to be mapped to $X_i$. And this is enough to define a map $S^{-1}A\to A[X_i\mid i\in I]$ if we leave $A$ fixed, i.e., $a\mapsto a$ for all $a\in A$. I'll let you finish the definition of the map from here.

With that definition it is more or less clear that the map is surjective and the kernel is precisely $\langle t_iX_i-1\mid i\in I\rangle$ since the only relation we are introducing is $t_iX_i=1$ from the fact that $t_it_i^{-1}=1$.