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 — Solution

2017 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2017 AMC 12B, 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.

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Concepts:algebraic manipulation substitution

Difficulty rating: 1130

Suppose that $x$ and $y$ are nonzero real numbers such that

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

What is the value of $\frac{x + 3y}{3x - y}?$

$-3$

$-1$

$1$

$2$

$3$

Solution:

The equation gives $3x + y = -2(x - 3y) = -2x + 6y,$ so $5x = 5y,$ meaning $x = y.$ Then $\frac{x + 3y}{3x - y} = \frac{y + 3y}{3y - y} = \frac{4y}{2y} = 2.$

Thus, the correct answer is D.

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