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Exercise 32.1

Exercise 32.1 Prove claim 2 here: given any two equations we define their sum to be the equation whose left hand side is the sum of the two left hand sides, and whose right hand side is the sum of the two right hand sides. Then you can replace either of the two equations by its sum with any multiple of the other without changing their implications.

Solution:

If both equations are true then any sum of the first with a multiple of the second will be true as well. Thus the only thing to worry about is that information could be lost in replacing the first equation by such a sum.
But you can go back and subtract the same multiple of the second from the new sum equation to get back the original equation that was replaced. So no information can possibly have been lost here and the implications of the original pair of equations is the same as that of the new pair.

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