AMC10 2002 B

AMC10 2002 B · Q12

AMC10 2002 B · Q12. It mainly tests Rational expressions, Manipulating equations.

For which of the following values of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$?

对于下列哪个$k$值，方程$\frac{x-1}{x-2}=\frac{x-k}{x-6}$没有$x$的解？

(A) 1 1

(B) 2 2

(D) 4 4

(E) 5 5

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) From the given equation we have $(x-1)(x-6)=(x-2)(x-k)$. This implies that $x^2-7x+6=x^2-(2+k)x+2k$, so $(k-5)x=2k-6$ and $x=\dfrac{2k-6}{k-5}$. Hence a value of $x$ satisfying the equation occurs unless $k=5$. Note that when $k=6$ there is also no solution for $x$, but this is not one of the answer choices.

（E）由已知方程可得 $(x-1)(x-6)=(x-2)(x-k)$。这意味着 $x^2-7x+6=x^2-(2+k)x+2k$，所以 $(k-5)x=2k-6$ 且 $x=\dfrac{2k-6}{k-5}$。因此，除非 $k=5$，否则存在满足该方程的 $x$ 值。注意当 $k=6$ 时，$x$ 也无解，但这不在给出的答案选项中。

Topics