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

[gmattesttaker2]

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

[[email protected]]

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

[gmattesttaker2]

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

[GMATGuruNY]

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.

[confused13]

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.

[GMATGuruNY]

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.
x² + y² = r² 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: x² + y² = 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 y≠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 y≠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.

[[email protected]]

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