Suppose that x and y are nonzero real numbers such that (3x + y)/(x - 3y) = -2. What is the value of (x + 3y)/(3x - y)?

2017 AMC 12B Problem 3

Attempt Problem 3 of the 2017 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Suppose that

x

and

y

are nonzero real numbers such that

\frac{3x + y}{x - 3y} = -2.

What is the value of

\frac{x + 3y}{3x - y}?

-3

-1

1

2

3

Clear the denominator:

3x + y = -2(x - 3y)

This simplifies to

x = y;

substitute into the target expression

The equation gives

3x + y = -2(x - 3y)

= -2x + 6y,

5x = 5y,

meaning

x = y.

Then

\frac{x + 3y}{3x - y} = \frac{y + 3y}{3y - y} = \frac{4y}{2y} = 2.

Thus, the correct answer is D.