If xy = -6, what is the value of \(\mathrm{xy(x + y)}\)? x - y = 5 xy^2 = 18...

Understanding the Question

We need to find the value of \(\mathrm{xy(x + y)}\).

Since we're given that \(\mathrm{xy = -6}\), this expression becomes:
\(\mathrm{xy(x + y) = -6(x + y)}\)

So the real question is: Can we determine a unique value for \(\mathrm{(x + y)}\)?

This is a "value" question, so we need sufficient information to find one specific numerical value for \(\mathrm{(x + y)}\), not a range of possible values.

Analyzing Statement 1

Statement 1: \(\mathrm{x - y = 5}\)

Now we have:
- \(\mathrm{xy = -6}\)
- \(\mathrm{x - y = 5}\)

Since \(\mathrm{xy = -6}\) (negative), x and y must have opposite signs. And since \(\mathrm{x - y = 5}\) (positive), we know that x must be positive and y must be negative.

Let's test if different pairs \(\mathrm{(x, y)}\) can satisfy both conditions:

Test Case 1: Let's try \(\mathrm{x = 3, y = -2}\)
- Check: \(\mathrm{xy = 3(-2) = -6}\) ✓
- Check: \(\mathrm{x - y = 3 - (-2) = 5}\) ✓
- Therefore: \(\mathrm{x + y = 3 + (-2) = 1}\)

Test Case 2: Let's try \(\mathrm{x = 2, y = -3}\)
- Check: \(\mathrm{xy = 2(-3) = -6}\) ✓
- Check: \(\mathrm{x - y = 2 - (-3) = 5}\) ✓
- Therefore: \(\mathrm{x + y = 2 + (-3) = -1}\)

We found two different pairs that satisfy both conditions, but they give different values for \(\mathrm{(x + y)}\): 1 and -1.

Since we cannot determine a unique value for \(\mathrm{(x + y)}\), we cannot determine a unique value for \(\mathrm{xy(x + y)}\).

Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: \(\mathrm{xy^2 = 18}\)

We have:
- \(\mathrm{xy = -6}\)
- \(\mathrm{xy^2 = 18}\)

Here's the key insight: Since \(\mathrm{xy^2 = (xy) \cdot y}\), we can write:
\(\mathrm{18 = (-6) \cdot y}\)

Solving for y:
\(\mathrm{y = 18 \div (-6) = -3}\)

Now that we know \(\mathrm{y = -3}\), we can find x:
\(\mathrm{xy = -6}\)
\(\mathrm{x(-3) = -6}\)
\(\mathrm{x = -6 \div (-3) = 2}\)

Therefore:
- \(\mathrm{x + y = 2 + (-3) = -1}\)
- \(\mathrm{xy(x + y) = -6(-1) = 6}\)

We get a unique value.

[STOP - Sufficient!]

Statement 2 is sufficient.

This eliminates choices C and E.

The Answer: B

Statement 2 alone gives us a unique value for \(\mathrm{xy(x + y)}\), while Statement 1 alone does not.

Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."