If a, b, and c are different constants, how many different numbers y

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 · Accepted 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

8Harshitsharma · Answer

If a ≠ 0, then y will have only one value.

But, if a = 0 then y can have infinitely many values.

Cannot be determined with given info.  Option E

Regor60 · Answer

The key is to recognize that when the question says "constants" it means that there is only one set of constants, not an infinite number of them.

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anchal02kansal · Answer

Hey, I'm unable to understand the answer. Ideally, if a=0, won't the whole equation collapse as it will make b=c, which is not possible because they are different constants. So shouldn't we eliminate a=0 case as a whole?

aniruddharora · Answer

Since the equasion is valid a*y + b = C, isn' t it assumed that a is not equal to zero, since b and c are different and hence the answer being 1.

Bunuel · Answer

Imagine a game where a, b, and c stand for some constants, but we don't know which ones. Next, a host writes an equation: ay + b = c and asks how many values of y can satisfy this equation. As shown in the solution, if a = 0, then there is no such y for ay + b = c to hold. If a is not 0, then there is one value of y such that a*y + b = c holds.

aniruddharora · Answer

Got it , I assumed that the equation must be satisfied not like for what value of y will it be satisfied

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EagerShip · Answer

if given a,b,c are constant value how can we assume a=0

Bunuel · Answer

A constant is simply a fixed value, it can still be 0. Since a, b, and c are different, a could be 0 as long as b and c are distinct nonzero values, for example, a = 0, b = 1, and c = 2.

egmat · Answer

Hi EagerShip,

I can see exactly where this is snagging you. You're reading the word  "constant"  as if it means "a specific, known value that can't be zero." That's the piece to fix.

What "constant" actually means:  it just tells you the letter stands for a  fixed number  - one unchanging value, as opposed to a variable that ranges. It does  not  tell you  what  that value is, and it does  not  rule out  0 . A constant can perfectly well equal  0 .

So look at the only restriction the question gives you: a, b, and c are  different  from one another. That's it. Nothing says a ≠  0 .

That means a =  0  is fully allowed, as long as b and c stay different from a and from each other - for example a =  0 , b =  2 , c =  5 . All three are different constants, and a happens to be  0 .

So we're not "assuming" a =  0  - we're recognizing it's a  possibility we can't rule out :

- If a ≠  0 : exactly  1  value of y works, namely y = (c - b) / a.
- If a =  0 : the equation becomes b = c, which is false, so  0  values of y work.

Since the count is  0 or 1  depending on whether a is zero, and the question never tells us, the answer is  E - cannot be determined .

Quick check to lock it in:  "k is a constant" - can k =  0 ? Yes. "m is a nonzero constant" - now can it? No. Notice the second one had to  say  "nonzero." Your question never did, so  0  stays on the table.

Answer: E

If a, b, and c are different constants, how many different numbers y

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If a, b, and c are different constants, how many different numbers y

Prep Toolkit

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