解く and 解答: Solving and the answer or reasoning
Compare solving an equation with an answer and the reasoning used to reach it. Preserve the separate shoelace-untying use of 解く and avoid treating every answer as correct.
The object tells you what is being done
解く includes solving and untying. The teacher's equation example reports the action of solving a problem on the board, while another example has a shoelace as its object.
「家に上がる前に靴ひもを解く。」 (I untie my shoelaces before entering the house.) The object untied before entering the house is a shoelace. This action does not become answering a test question.
An answer can be discussed with its reasoning
解答 refers to an answer or solution. The example of friends checking their answers after a test reports an exchange about answers, without establishing that every answer was correct.
「先生は正解だけでなく、解答に至る考え方も説明した。」 (The teacher explained not only the correct answer but also the reasoning used to reach it.) The explanation includes both the correct answer and the reasoning used to reach it. Answer content is not restricted to a single number or final result.
Separate the action from answer content
Solving an equation and discussing reasoning toward an answer have connected but different focuses. The teacher solving an equation and the teacher explaining reasoning are separate examples, not a reported sequence in one lesson.
With the verb, inspect the equation or shoelace being acted on. With the answer, inspect the answer or solution content, without treating the existence of an answer as a guarantee of accuracy.
What do solving an equation with 解く and 解答に至る考え方 focus on?
Show the answer and why
The first focuses on solving the equation; the second on the reasoning that leads to an answer. A separate 解く example involves untying a shoelace, so its object matters. Checking answers with someone also does not establish that every answer was correct.