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2017 AMC 10B Problem 4

2017 AMC 10B Problem 4AMC 10 Step by Step

Name: Date: September 13, 2026

Algebraic Manipulation · difficulty 1 of 5 · about 2 min · position P1-10

Suppose that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$ . What is the value of $\frac{x+3y}{3x-y}$ ?

A)
$-3$
B)
$-1$
C)
$1$
D)
$2$
E)
$3$

Answer

Key insight

Cross-multiplying the given equation collapses to x = y, after which the target fraction is 4x/2x = 2.

Solution

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 $\boxed{\textbf{(D)}\ 2}$ .

Why this works

A single equation in two unknowns cannot pin down $x$ and $y$ , but it can pin down their ratio, and a homogeneous expression (every term of the same degree) depends only on that ratio. Solve for the relation first, then substitute; the target fraction becomes trivial.

Alternative approach

Since only the ratio matters, set $x = y = 1$ after finding $x = y$ (check: $\frac{3+1}{1-3} = -2$ ). Then $\frac{1+3}{3-1} = 2$ .

The trap

Trying to compute the target expression by manipulating the given fraction directly, instead of first finding the relation between x and y.

Common mistakes

Trying to compute the target expression by manipulating the given fraction directly, instead of first finding the relation between x and y.
A sign slip when distributing $-2$ , producing $x = -y$ or similar and a wrong choice.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer