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If y is not equal to 1, is x = 1?

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Source: — Data Sufficiency |
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by 800_or_bust » Mon Jun 27, 2016 9:45 am
prata wrote:If y is not equal to 1, is x = 1?

1. x2 + y2 = 1

2. y = 1 - x
(1) Not sufficient. x could be zero, and y=-1. Or x could be +/-1, in which case y=0. Or x^2 = y^2 = 1/2. Etc...

(2) Not sufficient. y could be any value, except 1. Thus x can be any value except 0.

(1) & (2) combined: This is sufficient. Combining the two statements:

x^2 + (1-x)^2 = 1

Then x^2 + (1 - 2x + x^2) = 1

2x^2 - 2x = 0

x(2x - 2) = 0. So x = 0 or x = 1, but x cannot equal 0 from (2) above. Hence, x = 1 and y = 0.

Answer is C.
Last edited by 800_or_bust on Mon Jun 27, 2016 9:53 am, edited 3 times in total.
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by Rich.C@EMPOWERgmat.com » Mon Jun 27, 2016 9:46 am
Hi prata,

This DS question can be solved by TESTing VALUES.

We're told that Y CANNOT be 1. We're asked if X = 1. This is a YES/NO question.

1) X^2 + Y^2 = 1

IF....
X = 1
Y = 0
Then the answer to the question is YES.

IF....
X = -1
Y = 0
Then the answer to the question is NO.
Fact 1 is INSUFFICIENT

2) Y = 1 - X

Here, you might find it helpful to rewrite the equation as...

X + Y = 1

IF....
X = 1
Y = 0
Then the answer to the question is YES.

IF....
X = -1
Y = 2
Then the answer to the question is NO.
Fact 2 is INSUFFICIENT

Combined, we know...
X^2 + Y^2 = 1
X + Y = 1

The only pair of values that will 'fit' both equations at the same time are 0 and 1. Since we're told that the Y CANNOT be 1, then it must be the 0. By extension, the X MUST be 1 and the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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by Brent@GMATPrepNow » Mon Jun 27, 2016 10:12 am
There's a nice discussion here with some other approaches: https://www.beatthegmat.com/if-y-is-not- ... 76846.html

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by prata » Wed Jun 29, 2016 7:53 am
Would it be safe to assume that while solving inequalities only integers have to be considered while testing values of x and y?
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by DavidG@VeritasPrep » Wed Jun 29, 2016 7:56 am
prata wrote:Would it be safe to assume that while solving inequalities only integers have to be considered while testing values of x and y?
Not safe at all! (Though, obviously, integers would be easier to use, so you'd likely start with those values.) If the question doesn't specify that you're talking about integers, and it's not a word problem involving entities that would logically take integer values (you wouldn't worry about 1/3 of a cat) you cannot assume that the variables can only assume integer values.
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by Rich.C@EMPOWERgmat.com » Wed Jun 29, 2016 8:56 am
Hi prata,

On Test Day, on most of the questions that you can solve by TESTing VALUES, sticking with positive integers will be enough for you to get to the correct answer. However, certain questions will 'hinge' on the idea that the variables could be negative, or 0 or a non-integer (and you'll almost certainly face some of these issues in Data Sufficiency). To score at the higher levels in the Quant section, you have to be flexible enough in your thinking to realize that there are OTHER options besides positive integers.

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by OptimusPrep » Thu Jun 30, 2016 7:25 pm
prata wrote:Would it be safe to assume that while solving inequalities only integers have to be considered while testing values of x and y?
On the GMAT, you would be given everything related to the question.
If not given, do not limit yourself. Especially in the DS questions, where some of the values might be fractions.
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by Matt@VeritasPrep » Fri Jul 08, 2016 2:34 pm
S1:

x² + y² = 1

If y = 0, then x = 1 or x = -1. If y ≠ 0, then x ≠ 1. NOT SUFFICIENT

S2:

y = 1 - x

If y = 0, then x = 1. If y ≠ 0, then x ≠ 1. NOT SUFFICIENT

S1 + S2

x² + y² = 1
x² + (1 - x)² = 1
x² + 1 - 2x + x² = 1
2x² - 2x = 0
x * (x - 1) = 0

So x = 0 or x = 1. But if x = 0, then y = 1, and we were told in the stem that y ≠ 1. So we must have x = 1, and we're set!
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