I know the difference in meaning between the two: できます being "to be able to"
or "can do", and わかります being "to understand how to" or "to know". I also know
a simple sentence structure for both: "Person は Noun が (できます/わかります)" However,
I can't understand which one to use in a sentence. Especially with the use of
dictionary form verbs and こと in place of the noun.

I am trying to learn about this grammar point for a test and I just don't
understand them. Here is an image of my notes so you see exactly what I'm
seeing:

![notes](https://i.stack.imgur.com/0j96p.jpg)
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question
grammar word-choice
When do I use できます as opposed to わかります in a sentence?
To keep things a little simple, I'll just extract one example of each case
from all the ones you have on your sheet.

> サリさんは日本語{にほんご}が分{わ}かります。

This is fairly straightforward. It says, "Sally _understands_ Japanese."

> サリさんは日本語{にほんご}ができます。

Here, we're saying Sally has an ability for Japanese. A very direct, slightly
clunky, translation might be something like, "Sally is Japanese-capable." You
might want to see [this answer about the implications of
`できる`](https://japanese.stackexchange.com/a/2719/119).

So, which would you choose? Well, just like when speaking English, saying
someone understands Japanese, and that one is capable of Japanese, convey
largely the same general idea, and in most daily conversation you probably
won't go wrong exchanging one for the other. The difference is a matter of
nuance. The first seems, to me, to lean towards the idea that if you speak
Japanese _to_ Sally, she'll understand you. The second seems to convey more
that Sally can speak Japanese back to you.

Hope that helps.
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