2017 AMC 10B 第 4 题

Question

设

x

和

y

是非零实数，且满足

\frac{3x+y}{x-3y}=-2\text{。}

求下式的值：

\frac{x+3y}{3x-y}\text{。}

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

将给定方程清除分母。

Clear the denominator in the given equation

这个方程很快会推出

x

与

y

的关系。

The equation quickly forces a relation between

x

and

y

由

\frac{3x+y}{x-3y}=-2\text{，}

两边同乘分母，得到

3x+y = 6y-2x\text{。}

整理可得

x=y

。

因此

\frac{x+3y}{3x-y} = \frac{x+3x}{3x-x}=2\text{。}

所以正确答案是 D。

Given that

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

we can multiply by the denominator to get

3x+y = 6y-2x.

Solving, we can see that

x=y.

Therefore,

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

Thus, the correct answer is D .

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