Math CentralQuandaries & Queries


Question from Johanna, a student:

For which integers a, b, c does a - b - c = a - (b - c)?


I would first subtract a from each side.

Does that help?

If you need more assistance write back.


Johanna wrote back

Penny, I am still confused. I really don't undertand what the question is asking. I also don't understand how the first equation can equal the other one because the second one has parenthesis. Wouldn't -(b-c) equal -b + c?

Exactly! So now you have

-b -c = -b + c

Using my first suggestiion you eliminated a by subtracting a from each side. With the equation you are left with how do you eliminate b?


Johanna wrote back again

Penny, if I subtract a on both sides, then add b to both sides, that will give me -c = c. I still don't understand what I am supposed to be doing. What does "For which integers mean"? I thought I was supposed to plug in numbers or something. I am so confused.

Don't be put off, you are doing fine. This abstract stuff is hard to get your head around at first.

You have -c = c. Add c to each side. What does the result tell you about c?


Johanna wrote back again

If I add c to each side, would it be 0 = c + c or 0 = 2c ? I'm not sure what the result is telling me. Is it that c = 0 ? Or that c is twice as much as a and b?

Since 2c = 0 it must be that c = 0.

So what you have done is show that if a - b - c = a - (b - c) then c = 0. You didn't determine anything about a and b just that c = 0.

Now what about going backwards. Suppose c = 0, what can you say about a - b - c and a - (b - c)?


Johanna wrote back again

If c = 0 then
a - b - 0 = a - (b - 0)
a - b = a - b

So the equation is true? Is that the answer?

You started with a - b - c = a - (b - c) and asked what does that tell us about a, b and c? The conclusion was c = 0.

Then you started with c = 0 and asked are a - b - c and a - (b - c) equal? The answer was yes.

The original question was "For which integers a, b, c does a - b - c = a - (b - c)?" The answer is that a and b can be any integers but c must be 0.


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