Hello! I was hoping someone could look over my answers? It's very much appreciated! :)

How is the bracketed noun clause used?
Subject, Direct object, Predicate noun, or Object of preposition.

1. I am optimistic enough to believe [that young people will respond to a challenge.] <-- I know it's not a PN. I think it probably would be a Direct Object.

Different directions: Identity each italicized subordinate clause as a noun clause, an adjective clause, or an adverb clause.

2. Washington played by the river [when he was young.] <--- I think it would be an adverb clause

3. Red was the color [she chose for the drapes.] <-- I'm thinking it would be an adjective clause? :)

I had previously gotten these ones wrong, so I corrected them & I think they're correct now :)

Before when I had done them, the answers I had were: #1-PN #2-Noun clause #3-also noun clause. Thanks again! Your kindness is very much appreciated! :)

Your new answers are correct.

8x+14y=87.40

X+y=22

Hello! I'll be happy to look over your answers and provide some guidance on how to determine the correct answers.

1. I am optimistic enough to believe [that young people will respond to a challenge.]

You're right in saying that it's not a Predicate Noun (PN). To determine whether it is a Subject, Direct Object, or Object of preposition, we need to identify the function of the entire noun clause "that young people will respond to a challenge." In this case, the noun clause is functioning as the direct object of the verb "believe." Therefore, the correct answer is Direct Object.

2. Washington played by the river [when he was young.]

To identify the function of the italicized subordinate clause, we need to determine its purpose in the sentence. This clause provides information about the time when Washington played by the river. Since it is modifying the verb "played," it is acting as an adverb clause.

3. Red was the color [she chose for the drapes.]

Similarly to the previous example, we need to identify the function of the italicized subordinate clause. In this case, the clause "she chose for the drapes" is providing additional information about the noun "color." It is acting as an adjective clause, modifying the noun "color."

To recap:

1. The bracketed noun clause "[that young people will respond to a challenge" is functioning as a Direct Object.

2. The italicized subordinate clause "[when he was young]" is acting as an adverb clause.

3. The italicized subordinate clause "[she chose for the drapes]" is acting as an adjective clause.

I hope this helps clarify the answers for you! If you have any more questions or need further assistance, feel free to ask.