M41-13

Question

If  a ,  b , and  c  are different constants, how many different numbers  y  are there such that  a*y + b = c ?

A. 0
B. 1
C. 2
D. Infinitely many
E. Cannot be determined from the given information

Bunuel · Answer

Official Solution:

If  a ,  b , and  c  are different constants, how many different numbers  y  are there such that  a*y + b = c ?

A. 0
B. 1
C. 2
D. Infinitely many
E. Cannot be determined from the given information

Given  c  and  b  are different numbers, then  c - b 
eq 0 , this means  a*y = c - b = some \ nonzero \ number .
 
If  a = 0 , then no value of  y  can satisfy this equation, as  0*y = nonzero \ number  has no solutions for  y .
 
If  a 
eq 0 , then only 1 value of  y  can satisfy this equation, since in this case  y = \frac{c - b}{a} .
 
Therefore, there is either one solution or no solution, depending on the value of  a .

Answer: E

Samé · Answer

Bunuel 
Please, Why if a#0 then only one value of y satisfy the equation ? It is not said that y is an integer.

Posted from my mobile device

Bunuel · Answer

How does it matter whether y is an integer? If a is not 0, then for constants a, b, and c, y equals (c - b)/a, which would be some specific value.

shubha4world · Answer

I think this is a  high-quality  question and I don't agree with the explanation. In the solution, "a" can't be 0, because, if a = 0, for any value of y, b will always be equal to c, which is not possible as a,b,c are different (given).
So, as a can't be 0, there is only 1 possible solution for y.
It can be determined.
(Pls correct me if I am wrong)

Bunuel · Answer

Yes, you’re wrong; you misunderstood the question.

Case 1: If a = 0, there is no solution (specifically because b does not equal c).
Case 2: If a ≠ 0, there is 1 solution.

phmahi1997 · Answer

Hi, I have seen similar types of questions with the same stem; the only difference is that a, b, and c are positive and different constants!

Correct me if I am wrong:
If a, b, c were positive constant, that means a is not equal to zero; hence it is producing only one solution for y!

And when it is mentioned different constant, that means we can assume any value including 0, meaning we're getting two other cases; hence, It cannot be derived from the existing information given in the stems.

Thanks for the high quality question. I was confused at first with the concept behind this one.

Link to the other question: https://gmatclub.com/forum/m41-429156.html

Bunuel · Answer

Yes, the fact that a, b, and c are distinct constants means that at most one of them can be 0.

ritumalani_ · Answer

I like the solution - it’s helpful. Good Question

rak08 · Answer

if y = 0 ,
then a (0) + b = c
so b = c, 
they are different numbers this invalidated the equation !

Bunuel , why such contradiction?
even if we say y = (b-c)/a --> if Y is zero, b has to be equal to a.

give me an example when y can be zero with all consonants being distinct.

also can i know the source for this ques?

Bunuel · Answer

You misunderstood both the question and the solution. The question never asks whether y equals 0 or not. It asks  how many different values of y  satisfy the equation a*y + b = c. That means we’re interested in the  number of solutions for y , not what those solutions are.

As explained, if a ≠ 0, then there is exactly one solution, whatever it might. If  a = 0 , then there is no solution. Since we don’t know the value of a, we cannot determine how many values y can take.

As for the source, all questions in  this subforum  are from GMAT Club Tests.

Confusion · Answer

I don’t quite agree with the solution. A,b and c are different. So a can never be 0 as c-b can never be 0 since they are different/distinct. And if a is 0 and b & c are different and not equal to each other, then the equation itself will not hold true

Confusion · Answer

A,b and c are different. So a can never be 0 as c-b can never be 0 since they are different/distinct. And if a is 0 and b & c are different and not equal to each other, then the equation itself will not hold true

Bunuel · Answer

That’s the point. We have the equation a*y + b = c.

If a = 0, then the equation becomes b = c, which is impossible since b ≠ c, so if a = 0, there is no solution. No y can satisfy ay + b = c. For example, if a = 0, b = 1, and c = 2, then we get 0*y + 1 = 2, and no y can satisfy this equation.

If a ≠ 0, then y = (c - b)/a, so one solution.

Since we don’t know whether a = 0 or a ≠ 0, we cannot determine the number of solutions. The answer is E.

Sss678 · Answer

I like the solution - it’s helpful.

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