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What is the value of x? (1) x+y=2 (2) xy=z2+1
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31 Oct 2014, 13:12
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44% (01:48) correct 56% (01:23) wrong based on 288 sessions
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What is the value of x? (1) \(x+y=2\) (2) \(xy\)\(=\)\(z^2\)\(+1\)
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What is the value of x? (1) x+y=2 (2) xy=z2+1
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Updated on: 10 Jul 2019, 04:43
Veritas Official ExplanationYou should quickly see that neither statement alone is sufficient, which should signify to you that you'll have to do some work on the C vs. E decision. And here, algebra is the way to roll up your sleeves and get to work. If you take the first equation, x+y=2, and set it up to substitute in the second, you'll have: x=2−y and xy=z2+1 Substituting, that's: y(2−y)=z2+1 Which works out to: 2y−y2=z2+1 If you then set this up as a quadratic in terms of y, you have: −z2=y2−2y+1 And then factor the quadratic: −z2=(y−1)2 Now it's time to take a look logically. Since the left hand side cannot be positive (z2 can't be negative, so once you apply that negative sign to whatever the result is it's either 0 or something negative) and the right hand side cannot be negative (whatever (y−1) is, when you square it it's either positive or 0), you can determine that y−1 must be 0, and therefore that y=1. Given that, plug back into the first equation to find that x=1, proving the two statements together to be sufficient.
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Originally posted by JusTLucK04 on 31 Oct 2014, 13:23.
Last edited by MikeScarn on 10 Jul 2019, 04:43, edited 1 time in total.
Veritas Official Solution




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Re: What is the value of x? (1) x+y=2 (2) xy=z2+1
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20 Nov 2014, 10:11
S1:
x + Y = 2....what are the options for X & Y...X,Y can be combinations of [1,1] [0,2] [3/2, 1/2] [1/3, 5/3] etc....hence insufficient.
S2:
XY = z^2 + 1 .....with this we can not say what is x....hence insufficient.
Combining S1 & S2 together lets us plugin those options of S1 and see...
First lets take the fractions....e.g. 3/2, 1/2......so 3/4 = Z^2+ 1...now Z^2 = 1/4 But Z^2 can't be negative...so 3/2,1/2 is ruled out. Similarly all fractions options can be ruled out.
Now lets take 0,2 or 2,0......so now the eqn becomes 0=Z^2+1....Z^2 can't be 1 so...this option is also ruled out.
So finally we are left out with only option [1,1]. Hence X=1. So Answer C.

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Re: What is the value of x? (1) x+y=2 (2) xy=z2+1
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16 Apr 2016, 08:18
JusTLucK04 wrote: What is the value of x?
(1) \(x+y=2\) (2) \(xy\)\(=\)\(z^2\)\(+1\) st1> Many values of x and y to get value 2. So clearly insuff St2> xy= z^2+1 so z^2= positive number and adding one will be positive number Z can be z= 1 or z= 1/8 and z= 1 but their sq will always be positive number xy = +ve so x= +ve and y= +ve or x=ve and y = ve combining both: x can be 1 and y can be 1 am I right?



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Re: What is the value of x? (1) x+y=2 (2) xy=z2+1
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31 Jul 2017, 02:41
JusTLucK04 wrote: You should quickly see that neither statement alone is sufficient, which should signify to you that you'll have to do some work on the C vs. E decision. And here, algebra is the way to roll up your sleeves and get to work. If you take the first equation, x+y=2, and set it up to substitute in the second, you'll have:
x=2−y and
xy=z2+1 Substituting, that's:
y(2−y)=z2+1 Which works out to:
2y−y2=z2+1 If you then set this up as a quadratic in terms of y, you have:
−z2=y2−2y+1 And then factor the quadratic:
−z2=(y−1)2 Now it's time to take a look logically. Since the left hand side cannot be positive (z2 can't be negative, so once you apply that negative sign to whatever the result is it's either 0 or something negative) and the right hand side cannot be negative (whatever (y−1) is, when you square it it's either positive or 0), you can determine that y−1 must be 0, and therefore that y=1. Given that, plug back into the first equation to find that x=1, proving the two statements together to be sufficient. Using the knowledge that both RHS and LHS are not negative in this equation derived by combining both statements , how do we get to know that y=1?



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Re: What is the value of x? (1) x+y=2 (2) xy=z2+1
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01 Aug 2017, 09:15
email2vm wrote: JusTLucK04 wrote: What is the value of x?
(1) \(x+y=2\) (2) \(xy\)\(=\)\(z^2\)\(+1\) st1> Many values of x and y to get value 2. So clearly insuff St2> xy= z^2+1 so z^2= positive number and adding one will be positive number Z can be z= 1 or z= 1/8 and z= 1 but their sq will always be positive number xy = +ve so x= +ve and y= +ve or x=ve and y = ve combining both: x can be 1 and y can be 1 am I right? Who said anything about the two being integers?
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Re: What is the value of x? (1) x+y=2 (2) xy=z2+1
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03 Aug 2017, 05:50
TheMastermind wrote: Using the knowledge that both RHS and LHS are not negative in this equation derived by combining both statements , how do we get to know that y=1?
Hi TheMastermind , Good question. I know that you are already aware of the fact that square of any integer is always positive. Now, you got a^2 =  b^2 This is only possible when we have both a and b = 0. Any non zero values of a and b will not make this expression hold true. Hence, I can conclude that y1=0. Hence, y = 1 or x = 1. Thus, C is correct.
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Re: What is the value of x? (1) x+y=2 (2) xy=z2+1
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01 Sep 2017, 07:54
JusTLucK04 wrote: What is the value of x?
(1) \(x+y=2\) (2) \(xy\)\(=\)\(z^2\)\(+1\) (1) + (2) y = 2  x x(2 x) = z^2 + 1 2x  x^2 = z^2 + 1  (z^2) = x^2  2x + 1  (z^2) = (x 1)^2 (a^2) = (b^2) IF AND ONLY IF a=b=0x = 1
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Re: What is the value of x? (1) x+y=2 (2) xy=z2+1
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20 Feb 2019, 07:13
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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What is the value of x? (1) x+y=2 (2) xy=z2+1
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22 Aug 2019, 02:19
JusTLucK04 wrote: What is the value of x?
(1) \(x+y=2\) (2) \(xy\)=\(z^2\)\(+1\) Asked: What is the value of x? (1) \(x+y=2\) x = 2y and y is unknown NOT SUFFICIENT (2) \(xy\)=\(z^2\)\(+1\) x =\(\frac{( z^2 + 1)}{y}\) and value of z and y are unknown NOT SUFFICIENT Combining (1) & (2) (1) \(x+y=2\) y = 2x (2) \(xy\)=\(z^2\)\(+1\) \(y = \frac{(z^2 + 1 )}{x}\) \(2x = \frac{(z^2 + 1 )}{x}\) \(2x  x^2 = z^2 + 1\) \(x^2  2x + 1 + z^2 = 0\) \((x1)^2 + z^2 = 0\) Since \((x1)^2\) \(\geq\) 0 & \(z^2\) \(\geq\) 0 x1=0 & z =0 x = 1 SUFFICIENT IMO C
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What is the value of x? (1) x+y=2 (2) xy=z2+1
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