J N wrote:
For integers a, b, and c, if ab = bc, then which of the following must also be true?
A) a = c
B) a^2b=bc^2
C) ac= 1
D) abc > bc
E) a + b + c = 0
Many students will incorrectly choose
A.
They'll arrive at this conclusion by taking ab = bc and dividing both sides by b to get a = c.
However, when they do this, they are forgetting that it's possible that b = 0, in which case they are unwittingly dividing both sides by zero.
Notice that one possible solution to the given equation is a=2, b=0 and c=3, in which case a does not equal b. So, we can eliminate A since we're looking for an answer choice that MUST ALWAYS be true.
Here's one way to handle the rest of the question . . .
Take ab = bc
Rearrange to get: ab - bc = 0
Factor to get: b(a - c) = 0
From this, we can conclude that b = 0
OR a - c = 0.
In other words b = 0
OR a = c
Now we'l check answer choice B.
Must it be true that a^2b = bc^2?
Yes!
Rearrange to get a^2b - bc^2 = 0
Factor: b(a^2 - c^2) = 0
Factor more: b(a + c)(a - c) = 0
Now we already know that b = 0
OR a = c
If b = 0, then b(a + c)(a - c)
must equal 0
If a = c, then b(a + c)(a - c)
must equal 0
Since both possible cases (b = 0 or a = c) result in answer choice B being true, the correct answer is
B
Cheers,
Brent
