Data Sufficiency

[GMAT math practice question]

What are the values of x+y and xy?

1) x + y + xy = -2
2) (1/x) + (1/y) = 1

Statement 1=> x + y + xy = -2
x + y = -2 - xy
OR
xy = -2 - x - y
The values of x and y are unknown, hence, statement 1 is NOT SUFFICIENT.

Statement 2=>
$$\left(\frac{1}{x}\right)+\left(\frac{1}{y}\right)=1$$
$$\frac{\left(y+x\right)}{xy}=1$$
$$xy=y+x\ and\ y+x=xy$$

Combining both statements;
Statement 1: x + y = -2 - xy --- (i)
Statement 2: xy = y + x ---- (ii)
x + y = -2 - xy
x + y = -2 - (y+x)
x + y = -2 - y - x
x + y + y + x = -2
2x + 2y = -2
2(x+y) = -2 --- (iii)
From statement 2, make x+y the subject of the formula
xy = y + x
-x - y = -xy
multiply both sides by -1
-1 (-x-y) = -1 (-xy)
x + y = xy substituting x+y into (iii)
2(x+y) = -2
2(xy) = -2
Divide both sides by 2, we have;
xy = -1
From statement 2, we know that xy and x+y are equal. Hence, xy=-1 and x+y=-1.
Therefore, both statements combined ARE SUFFICIENT.

Answer = option C

=>

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Since we have 2 variables (x and y) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since 1/x + 1/y = 1 from condition 2), we have y + x = xy by multiplying both sides of the equation by xy, which rearranges to get xy - (x+y) = 0.
Since xy + (x+y) = -2 from condition 1), we have xy - (x+y) + xy + (x+y) = 0 + -2 by adding the two equations. Then 2xy = -2 or xy = -1.
Then we have x+y=-1.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.