JusTLucK04 wrote:
What is the value of x?
(1) \(x+y=2\)
(2) \(xy\)\(=\)\(z^2\)\(+1\)
Target question: What is the value of x? Statement 1: x + y = 2 We can see that x can have infinitely many values.
Statement 1 is NOT SUFFICIENT
For example, we could have
x = 1 and y = 1 OR we could have
x = 0 and y = 2
Statement 2: xy = z² + 1Statement 2 is clearly NOT SUFFICIENT
For example, we could have
x = 1, y = 2 and z = 1 OR we could have
x = 2, y = 0.5 and z = 0
Statements 1 and 2 combined Take statement 1, x + y = 2, and rewrite as
y = 2 - xTake statement 2, xy = z² + 1, and replace
y with
2 - x to get: x(
2 - x) = z² + 1
Expand to get: 2x - x² = z² + 1
Multiply both sides by -1 to get: -2x + x² = -(z²) - 1
Rearrange to get: x² - 2x = -(z²) - 1
Add 1 to both sides to get: x² - 2x + 1 = -(z²)
Factor left side to get: (x - 1)² = -(z²)
IMPORTANT:
(x - 1)² is GREATER than
or equal to 0 for ALL values of x
-(z²) is LESS than
or equal to 0 for ALL values of z
So, the only way that the equation, (x - 1)² = -(z²), can be true is when (x - 1)² and -(z²)
both equal 0If (x - 1)² = 0, then it MUST be the case that
x = 1 Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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