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Good question, because it focuses on a major weakness in the
teaching of math in this country.
What is algebra? It is a formal, systematic method of reasoning.
You start with a relationship (F=ma, or its differential, vector
equivalent) and deduce consequences of that relationship. Using algebra,
and obeying its rules, guarantees that the result is logically equivalent
to the starting point. In this way you discover that a huge class of
physical phenomena is consistent with that relationship. It is not that
nature is obeying the rules of algebra, it is that your conclusions are
correctly obtained if you correctly apply the rules of algebra.
Sometimes, of course, you reason correctly and obtain a result
that is different from the physical result. When that exciting event
occurs you are guaranteed that your starting principle was incorrect.
High school algebra teachers, many of them, have never learned
that algebra is nothing but a system for reasoning.