Are these fractions all equal?Are the following expressions all equal to one another?2ab+cy2aby+cy2ay⋅(2by)+cy2ab+cy=2aby+cy=2ay×2by+cy

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Cellamaran78 · Accepted Answer

By starting with the first, and applying the distributive law, we get the second:2ab+cy=(2ab+c)1y=2ab1y+c1y=2aby+cyThe last one is not equal to the other two, because2ay⋅2by+cy=2a⋅2by⋅y+cy=4aby2+cy,and 4aby2+cy is (usually) not equal to 2aby+cy. Notice that we used the fact thatab⋅cd=acbd.So the first two are equal, but not the third.

reclaimxww · Accepted Answer

Eivand&#039;s answer is complete - but I want to talk about a related idea. How does one know if something is true?Often, proving that something is true is incredibly complicated. To prove something false, one needs only a counterexample. But the first step should always be to experiment with the idea. In this case, this probably means making a few fractions of the forms that you have described, and testing to see if they are equal.After a few tests for the first 2 cases, you might convince yourself that they are true (this does not, of course, constitute a proof!). But the same heuristic approach would quickly destroy the third: suppose we leta=1,b=2,c=3,y=5.Then 2ay⋅2by+cy=25⋅45+35=825=1525=2325&amp;lt;1On the other hand, 2ab+cy=4+35&amp;gt;1. Almost any trial would yield a contradiction. I think that this style of patient problem-solving and analysis is under-emphasized and under-utilized, because it provides great insight on many questions.Reasonable to add?

Are these fractions all equal? Are the following expressions

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