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Based on the wording of the question and the fact

EDIT: Now that the OP is has edited his question to make clearer what he wants as an undergradanswer, I'm guessing he's looking for a relationship between weak equivalences and homotopy equivalences, not a way to do homotopy theory without mentioning homotopies. Something like ``every homotopy equivalence in $Ho(Top)$ comes from a unique weak equivalence in $Top$.'' Of course, this is nowhere near trueremoving speculation about what he wanted. The question links to an article answer is: yes, you can characterize homotopy equivalences as the maps which describes how become isomorphisms after applying the localization functor to define a homotopy in a model category using Path objects and Cylinder objectsinvert the weak equivalences. Once This answer doesn't require you have a to "use the notion of homotopy, " because it's easy to get homotopy equivalences in the usual way ($fg \simeq id_X, gf\simeq id_Y$)part of a much more general framework.

If you want

Here is a way to get at the homotopy equivalences directly from the weak equivalences, then I recommend you read this nice article on localization. One of the best reasons for studying model categories is that they let you get your hands on the maps (weak equivalences) which build homotopy equivalences. Many of the axioms for a model category are there to get around set-theoretic issues that arise when you try to localize a category with respect to an arbitrary class of morphisms. It turns out you need to localize at a class of morphisms which admits a calculus of fractions, and the class $W$ of weak equivalences does. If you read the article, you'll see how to construct $C[W^{-1}]$ in general, and you can then specialize to the case where $C=Top_*$ and $W$ is the class of weak equivalences.

Unfortunately, for computation, this highly general approach is not as good as just using the Path and Cylinder objects mentioned in the nLab article the OP links to. That's why most people who study model categories use those instead of this general framework: because a model category is more than just a category with a nice class $W$--it's a category where you can really get your hands on everything and compute!Still, I felt like the question was really asking about this general framework and not what's best for computation.
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Based on the wording of the question and the fact that the OP is an undergrad, I'm guessing he's looking for a relationship between weak equivalences and homotopy equivalences, not a way to do homotopy theory without mentioning homotopies. Something like ``every homotopy equivalence in $Ho(Top)$ comes from a unique weak equivalence in $Top$.'' Of course, this is nowhere near true. The question links to an article which describes how to define a homotopy in a model category using Path objects and Cylinder objects. Once you have a notion of homotopy, it's easy to get homotopy equivalences in the usual way ($fg \simeq id_X, gf\simeq id_Y$).

If you want a way to get at the homotopy equivalences directly from the weak equivalences, then I recommend you read this article on localization. One of the best reasons for studying model categories is that they let you get your hands on the maps (weak equivalences) which build homotopy equivalences. Many of the axioms for a model category are there to get around set-theoretic issues that arise when you try to localize a category with respect to an arbitrary class of morphisms. It turns out you need to localize at a class of morphisms which admits a calculus of fractions, and the class $W$ of weak equivalences does. If you read the article, you'll see how to construct $C[W^{-1}]$ in general, and you can then specialize to the case where $C=Top_*$ and $W$ is the class of weak equivalences.

Unfortunately, for computation, this highly general approach is not as good as just using the Path and Cylinder objects. That's why most people who study model categories use those instead of this general framework: because a model category is more than just a category with a nice class $W$--it's a category where you can really get your hands on everything and compute! Still, I felt like the question was really asking about this general framework and not what's best for computation.