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## Main Question or Discussion Point

I am currently working my way though Calculus by Tom Apostol. One of the really early proofs ask the reader to prove: a(b-c)=ab-ac. Here is what I did, I let x=b-c which by the definition of subtraction equals x+c=b. Substituting that value into the right hand side I got a((x+c)-c)=a(x+(c-c))=a(x+0)=ax.

I then plugged the exact same value into the right hand side getting a(x+c)-ac=(ax+ac)-ac=ax+(ac-ac)=ax.

Is this sufficient as a proof? In a proof that I looked up the author of the proof instead let part of the left hand side =x and part of the right hand side equal y then showed that x=y. Is that the way I should have approached this problem?

I then plugged the exact same value into the right hand side getting a(x+c)-ac=(ax+ac)-ac=ax+(ac-ac)=ax.

Is this sufficient as a proof? In a proof that I looked up the author of the proof instead let part of the left hand side =x and part of the right hand side equal y then showed that x=y. Is that the way I should have approached this problem?