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

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Bunuel
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If a, b, and c are constants, how many different numbers y are there [#permalink] New post   10 Nov 2022, 12:13
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If \(a\), \(b\), and \(c\) are constants, how many different numbers \(y\) are there such that \(a*y + b = c\)?

(1) \(c >b\)

(2) \(a > 1\)


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B
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chetan2u
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   10 Nov 2022, 21:21
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Bunuel wrote:
How many different numbers y are there such that A*y + B = C ? (A, B, and C are constants)

(1) C > B
(2) A > 1



We are looking for number of values that y can take and not for specific value of y.

Since we are dealing with three constants, A, B and C, so equation Ay+B=C tells us that y should also have just one value as there is no INEQUALITY or VARIABLES we are dealing with.

However, if A is 0, y can be any value as Ay will also be 0.

(1) C > B
Does not matter as y can be fraction, integer or decimal to satisfy the equation.
Insufficient

(2) A > 1
So, we have only one value for y.
Sufficient


B
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Bunuel
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   11 Nov 2022, 03:09
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chetan2u wrote:
Bunuel wrote:
How many different numbers y are there such that A*y + B = C ? (A, B, and C are constants)

(1) C > B
(2) A > 1



We are looking for number of values that y can take and not for specific value of y.

Since we are dealing with three constants, A, B and C, so equation Ay+B=C tells us that y should also have just one value as there is no INEQUALITY or VARIABLES we are dealing with.

However, if A is 0, y can be any value as Ay will also be 0.

(1) C > B
Does not matter as y can be fraction, integer or decimal to satisfy the equation.
Insufficient

(2) A > 1
So, we have only one value for y.
Sufficient


B


The analysis for (1) is not precise. For (1), if A = 0, then no value of y can satisfy the equation. So, y cannot take any value.
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   11 Nov 2022, 03:10
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Bunuel wrote:
How many different numbers y are there such that A*y + B = C ? (A, B, and C are constants)

(1) C > B
(2) A > 1


This is a tricky algebra question, so I recommend studying it carefully.
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chetan2u
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   11 Nov 2022, 04:04
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Bunuel wrote:
chetan2u wrote:
Bunuel wrote:
How many different numbers y are there such that A*y + B = C ? (A, B, and C are constants)

(1) C > B
(2) A > 1



We are looking for number of values that y can take and not for specific value of y.

Since we are dealing with three constants, A, B and C, so equation Ay+B=C tells us that y should also have just one value as there is no INEQUALITY or VARIABLES we are dealing with.

However, if A is 0, y can be any value as Ay will also be 0.

(1) C > B
Does not matter as y can be fraction, integer or decimal to satisfy the equation.
Insufficient

(2) A > 1
So, we have only one value for y.
Sufficient


B


The analysis for (1) is not precise. For (1), if A = 0, then no value of y can satisfy the equation. So, y cannot take any value.


Yes, you are correct. A=0 will give C=B, while statement I says that C>B. Thus, A=0 will not be valid for statement 1.
y=(C-B)/A
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   11 Nov 2022, 07:55
IMO B.

According to the question.
Ay+B=C
Ay=C-B

Case I
C>B
then we know that RHS>0
If A!=0, then 1 solution
if A=0, then 0 solutions

Case II
A>1
B-C will take a definitive value and y=(B-C)/A

Alternatively y=(B-C)/A, for y to have a value, A cannot be equal to 0
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   09 Dec 2022, 09:41
Bunuel wrote:
Bunuel wrote:
How many different numbers y are there such that A*y + B = C ? (A, B, and C are constants)

(1) C > B
(2) A > 1


This is a tricky algebra question, so I recommend studying it carefully.


Will you provide a detailed solution and analysis?

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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   11 Dec 2022, 11:08
chetan2u wrote:
Bunuel wrote:
How many different numbers y are there such that A*y + B = C ? (A, B, and C are constants)

(1) C > B
(2) A > 1



We are looking for number of values that y can take and not for specific value of y.

Since we are dealing with three constants, A, B and C, so equation Ay+B=C tells us that y should also have just one value as there is no INEQUALITY or VARIABLES we are dealing with.

However, if A is 0, y can be any value as Ay will also be 0.

(1) C > B
Does not matter as y can be fraction, integer or decimal to satisfy the equation.
Insufficient

(2) A > 1
So, we have only one value for y.
Sufficient


B



Bunuel
Can you please explain -
A> 1 signifies A can take any values greater than 1.
And if that’s the case then y will have multiple values based on A (and also B and C because even though the value B and C ate constant, their value have not been defined )
?

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Bunuel
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   25 Apr 2023, 04:51
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Official Solution:


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

(1) \(c >b\)

Given \(c - b > 0\), this means \(a*y = some \ positive \ number\).

If \(a = 0\), then no value of \(y\) can satisfy this equation, as \(0*y = positive\ number\) has no solutions for \(y\).

If \(a ≠ 0\), then only 1 value of \(y\) can satisfy this equation, since in this case \(y = \frac{c - b}{a}\).

Not sufficient.

(2) \(a > 1\)

\(y = \frac{c - b}{a} = \frac{c - b}{nonzero \ number}\).

As a result, regardless of the value of \(c - b\), the equation \(a*y = c - b\) will always have only one solution, given by: \(y =\frac{c - b}{nonzero \ number}\).

Sufficient.


Answer: B
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink] New post   18 Oct 2023, 05:03
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Re: If a, b, and c are constants, how many different numbers y are there [#permalink]
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