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

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pm; Thanked: 11 times; Followed by:8 members

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

by gmattesttaker2 » Tue Jun 03, 2014 5:37 pm

Hello,

Can you please tell me if where I am going wrong here? I am getting E

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

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

OA: C

1) x^2 = 1 - y^2
y = 0 => x^2 = 1 => x = +/- 1 - In-suff

2) y = 1 - x
=> x = 1 - y
y = 0 => x = 1
y = 2 => x = -1

In-suff

1 and 2:
x^2 + y^2 = 1 and y = 1 - x
=> y^2 = 1 - x^2
=> y^2 = (1+x)(1-x)
=> y^2 = (1+x)y
=> y = 1+x
=> x = y - 1
y = 2 => x = 1
y = 0 => x = -1

Hence, in-suff.

Thanks a lot,
Sri

Quote

Source: Beat The GMAT — Data Sufficiency | Tue Jun 03, 2014 5:37 pm

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by [email protected] » Tue Jun 03, 2014 5:49 pm

Hi Sri,

You made a mistake when combining the equations; after substituting, you have to square the ENTIRE term:

X^2 + Y^2 = 1 and Y = 1 - X

Substituting in, we get...

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

The above is where you made the mistake (you wrote it as X^2 + 1 - X^2 = 1). From here, the math becomes....

X^2 + [1 - 2X + X^2] = 1

2X^2 - 2X + 1 = 1

2X^2 - 2X = 0

2X(X - 1) = 0

It would appear that X could be 0 or 1, HOWEVER... we're told in the prompt that Y CANNOT = 1.

With the equation: Y = 1 - X, since Y cannot be 1, X CANNOT be 0. This eliminates one of the above 2 options.

X can only = 1. The answer to the question is YES and it is the ONLY answer.
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

Quote

gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pm; Thanked: 11 times; Followed by:8 members

by gmattesttaker2 » Wed Jun 04, 2014 12:47 am

[email protected] wrote:Hi Sri,

You made a mistake when combining the equations; after substituting, you have to square the ENTIRE term:

X^2 + Y^2 = 1 and Y = 1 - X

Substituting in, we get...

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

The above is where you made the mistake (you wrote it as X^2 + 1 - X^2 = 1). From here, the math becomes....

X^2 + [1 - 2X + X^2] = 1

2X^2 - 2X + 1 = 1

2X^2 - 2X = 0

2X(X - 1) = 0

It would appear that X could be 0 or 1, HOWEVER... we're told in the prompt that Y CANNOT = 1.

With the equation: Y = 1 - X, since Y cannot be 1, X CANNOT be 0. This eliminates one of the above 2 options.

X can only = 1. The answer to the question is YES and it is the ONLY answer.
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Hello Rich,

Thanks for your prompt reply. I first simplified X^2 + Y^2 = 1

i.e. Y^2 = 1 - X^2
=> Y^2 = (1+X)(1-X)

Now I plugin Y = 1 - X
=> Y^2 = (1+X)Y

and then I cancel out Y
=> Y = 1+X
=> X = Y-1

Is this correct?

Thanks a lot for your help.

Regards,
Sri

Quote

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Wed Jun 04, 2014 2:07 am

gmattesttaker2 wrote: I first simplified X^2 + Y^2 = 1

i.e. Y^2 = 1 - X^2
=> Y^2 = (1+X)(1-X)

Now I plugin Y = 1 - X
=> Y^2 = (1+X)Y

and then I cancel out Y
=> Y = 1+X
=> X = Y-1

The error is the step in red.
To cancel out Y is to divide both sides of the equation by Y.
VERY IMPORTANT: Never divide by a variable if it's possible that the variable is equal to 0.
Here, it is possible that Y=0.
Thus, we cannot cancel out Y by dividing both sides of the equation by Y.

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confused13: Senior | Next Rank: 100 Posts; Posts: 97; Joined: Tue Dec 17, 2013 3:29 am; Thanked: 8 times; Followed by:2 members

by confused13 » Wed Jun 04, 2014 3:13 am

[email protected] wrote:Hi Sri,

You made a mistake when combining the equations; after substituting, you have to square the ENTIRE term:

X^2 + Y^2 = 1 and Y = 1 - X

Substituting in, we get...

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

The above is where you made the mistake (you wrote it as X^2 + 1 - X^2 = 1). From here, the math becomes....

X^2 + [1 - 2X + X^2] = 1

2X^2 - 2X + 1 = 1

2X^2 - 2X = 0

2X(X - 1) = 0

It would appear that X could be 0 or 1, HOWEVER... we're told in the prompt that Y CANNOT = 1.

With the equation: Y = 1 - X, since Y cannot be 1, X CANNOT be 0. This eliminates one of the above 2 options.

X can only = 1. The answer to the question is YES and it is the ONLY answer.
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

The GMAT world must look like kindergarten to you.
I reached to the final line as well

2X(X - 1) = 0

and based on this I deduced that both are not sufficient, what a horrible mistake.

Quote

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Wed Jun 04, 2014 10:49 am

gmattesttaker2 wrote:Hello,

Can you please tell me if where I am going wrong here? I am getting E

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

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

A very fast approach is to GRAPH the two statements.
xÂ² + yÂ² = rÂ² is the equation for a circle that is centered at the origin and has a radius of r.
y = mx + b is the equation of a line with a slope of m and a y-intercept of b.

Statement 1: xÂ² + yÂ² = 1
This is the equation for a circle that is centered at the origin and has a radius of 1:

The question prompt indicates that yâ‰ 1.
Thus, (x,y) can be any point on the circle other than (0,1).
Since it's possible that (x,y) = (1,0) or that (x.y) = any other point on the circle other than (0,1), INSUFFICIENT.

Statement 2: y = 1-x
Rephrased in the form of y = mx + b:
y = -x + 1.
This is the equation of a line with a slope of -1 and a y-intercept of 1:

The question prompt indicates that yâ‰ 1.
Thus, (x,y) can be any point on the line other than (0,1).
Since it's possible that (x,y) = (1,0) or that (x.y) = any other point on the line other than (0,1), INSUFFICIENT.

Statements combined:
Overlaying the graphs, we get;

Of the 2 points of intersection, only (1,0) is viable.
Thus, x=1.
SUFFICIENT.

The correct answer is C.

Quote

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by [email protected] » Wed Jun 04, 2014 5:01 pm

HI confused13,

Any time a GMAT question includes a tiny piece of "random" information, I'll think about WHY I was given that information. In this prompt, we're told that Y cannot = 1, but WHY were we told that? What role does that information play in the question?

In this prompt, that information barely affects Fact 1 (it just tells us one value that Y cannot equal), but in Fact 2 the information has a greater impact (since Y cannot be 1, X cannot be 0).

With enough practice, you'll come to find that the little things are essential to answering the question, so make sure to note them in your work and think about how they mesh with the other information that you have.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]