After my opinion it is not true: if you are good at mathematics you must be good at chemistry and physics.
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Algebra is for solving equations. It's the entire purpose of Algebra. You'll use Algebra in Biology, Chemistry, and Finance. Study hard and it will make your other math & science classes easier.
Yes, but as you get more into chemistry at higher levels, algebra will not be of any use to you(i.e organic chemistry, inorganic chemistry, etc).
Being able to use algebra is essential to studying physics. Being good with algebra can make studying physics easier than it would be if you were not good with algebra. However, being good with algebra will not ensure that you will find studying physics to be easy.
I was in Chemistry last year and we used something called stoichiometry, which uses both algebra and geometry.
Some jobs that need Algebra are chemistry.