Data Sufficiency

Hi,

Request help with understanding of the following question.Thanks..

x + y = ?

1.x^2 + y^2 = 5

2.xy = 2

shivanigs wrote:Hi,

Request help with understanding of the following question.Thanks..

x + y = ?

1.x^2 + y^2 = 5

2.xy = 2

It should be clear that neither statement is sufficient on its own, but you can pick numbers to verify this:

Statement 1: let x=0, so y= +-√5. So, x+y could be √5 or -√5. INSUFFICIENT.

Statement 2: Let x=1, so y=2, and x+y=3. Or, let x=4, so y=0.5, so x+y=4.5. INSUFFICIENT.

Statements 1&2 combined: You can use both equations to solve for possible values of x and y, but you might take a few seconds to see if you can figure out any possible values just by guessing and checking. x=1 and y=2 is a natural place to start for the second statement, because 2 and 1 are the only integer factors of 2. Plugging these values into statement 1 verifies that it is also a solution of x^2+y^2=5. So, in this case, x+y=3. Notice that x=-1 and y=-2 also solves both equations, but in this case x+y=-3. Thus, the answer is E.

If you don't see any possible solutions right away, you'll probably have to just solve it mathematically:

1. x^2+y^2=5, and y=2/x (Solve statement 2 for y)

2. x^2+(2/x)^2=5 (Substitution)

3. x^2+4/x^2=5

4. x^4+4=5x^2 (Multiplying both sides of the equation by x^2)

5. x^4-5x^2+4=0 (Rearranging equation into standard form)

6. (x^2-4)(x^2-1)=0 (Factoring)

7. x^2=4 or x^2=1, so x=+-2, or x=+-1

Plug these values back into either equation to get the four solutions:

x=1, y=2; x=2, y=1; x=-2, y=-1; x=-1, y=-2

Clearly 3 and -3 are both possible values for x+y, so again the answer is

E

Hi Shivani,

Using x^2 + y^2 = 5

and

xy = 2,

we can safely say that x^2 + y^2 + 2xy = 9,

or, (x+y)^2 = 9,

Using the result (x+y)^2 = 9, it can be deduced that (x+y) would be either +3 or -3.

Any additional information on the nature of x and y would have helped in arriving at the exact value of (x+y).

I guess this should've helped you.

Good Luck.

shivanigs wrote:Hi,

Request help with understanding of the following question.Thanks..

x + y = ?

1.x^2 + y^2 = 5

2.xy = 2

statement 1:

Pick the numbers
let x² = 4 , so x = ±2
Since x² + y² = 5
=> 4 + y² = 5
=> y² = 1
=> y = ±1

for x= 2, y = 1
x + y = 3
for x = -2 , y = -1
x + y = -3

We got two different values of x+y, hence the statement is insufficient

Statement 2:
xy = 2
if x = 2 , y= 1 and x+y = 3
if x = -2, y = -1 and x+y = -3

We got two different values of x+y, hence the statement is insufficient

combining both the statements

x² + y² = 5
add 2xy to both the sides

x² + y² + 2xy = 5 + 2xy
(x+y)² = 5*4 = 9 = (±3)²
=> x+y = ±3

insufficient

The correct answer is E