2017 AMC 10B Problem 4

One equation with two letters cannot tell us $x$ and $y$ separately, but it can tell us how they are related. So find that relation first.

1. Multiply both sides of the given equation by $x - 3y$ to clear the fraction:
$$3x + y = -2(x - 3y) = -2x + 6y.$$
2. Move the $x$ terms to the left and the $y$ terms to the right: $3x + 2x = 6y - y$, so $5x = 5y$, which means $x = y$.
3. Replace every $y$ in the asked fraction with $x$:
$$\frac{x + 3y}{3x - y} = \frac{x + 3x}{3x - x} = \frac{4x}{2x} = 2.$$
We may cancel the $x$ because the problem says $x$ is not zero.

The answer is $\boxed{\textbf{(D)}\ 2}$.

In both fractions every term is a number times $x$ or a number times $y$. Such a fraction does not change when $x$ and $y$ are both doubled or both halved, so it depends only on how $x$ and $y$ compare. The given equation tells us exactly that ($x = y$), and the rest is substitution.

Once you know $x = y$, pick the easiest numbers: $x = y = 1$. Check the given equation: $\frac{3 + 1}{1 - 3} = \frac{4}{-2} = -2$. Then the asked fraction is $\frac{1 + 3}{3 - 1} = \frac{4}{2} = 2$.

Clear the denominator in the given equation:
$$3x + y = -2(x - 3y) = -2x + 6y.$$
Collecting terms gives $5x = 5y$, so $x = y$.

Now substitute $y = x$ into the desired expression:
$$\frac{x + 3y}{3x - y} = \frac{x + 3x}{3x - x} = \frac{4x}{2x} = 2.$$
Because $x$ is nonzero, the division is legitimate.

The answer is $\boxed{\textbf{(D)}\ 2}$.