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In a problem my teacher did the following:

5a - 2b = 3m  5b - 2c = 3n    + ----------------- 5a - 2c + 3b = 3m + 3n 

I tried solving for b in the first equation and then plugging it in to the second but could not get the same result.

When is adding full equations like this valid?

Thanks!

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    "(It's expressed differently on the two sides, but it is the same quantity, after all.) " was exactly what I was looking for! Thank you so much!2010-10-29

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If A = B and C = D then A+C = B+D.

More generally, if $f$ is any function, then $x = y$ implies $f(x) = f(y)$.

In this case we have used the function f(x) = x + C. to go from A = B to A + C = B + C as well as g(x) = B + x to go from C = D to B + C = B + D. Then we paste the two equations together (transitivity) to get A + C = B + D.