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655-705 (Hard)|   Algebra|      
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If y ≠ 1, is x = 1 ? (1) x^2 + y^2 = 1 (2) y = 1 - x 24 Jul 2023, 08:09
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chuchosol03
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If y≠1, is x=1?

(1) \(X^2 + Y^2 = 1\)
(2) \(y = 1 - X\)

Hi Guys! I need your help!

My logic is following:
(1) not sufficient equation is virtually a circle around (0,0) point with the radius 1. If y ≠ 1, than x can be almost anything within following limits [-1;0) & (0;1]
(2) not sufficient: consider y = 0 => x=1; y = 3 => x=-2

(1)+(2) Substitute Y from (2) in (1):
\(X^2 + (1-X)^2=1\)
...
\(2X (X-1) = 0\)

Hense X= 0 or X = 1.

Not sufficient?

PLS Help! :)

The point is that x=0 is not a valid solution. For x = 0, from y = 1 - x it follows that y = 1 but we are told in the stem that y ≠ 1. Thus x can only be 1. Sufficient.

Answer: C.

Does this make sense?

What about y= -1 here?
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When we combine the statements, if y = -1, then from y = 1 - x, it would follow that x = 2. However, y = -1 and x = 2 does not satisfy x^2 + y^2 = 1. Therefore, y = -1 is not a solution for x^2 + y^2 = 1 and y = 1 - x.

Hope it helps.
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If y ≠ 1, is x = 1 ? (1) x^2 + y^2 = 1 (2) y = 1 - x 10 Dec 2025, 01:59
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