Question

对下列哪个 $k$ 值，方程没有 $x$ 的解？ $\dfrac{x - 1}{x - 2} = \dfrac{x - k}{x - 6}$

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

$1$

$2$

$3$

$4$

$5$

答案：E

知识点：分式方程一次方程

难度评级：1370

解答：

交叉相乘得 $(x - 1)(x - 6) = (x - 2)(x - k),$ 展开为 $\begin{aligned} x^2 - 7x + 6 &= x^2 - (2 + k)x \\ &\quad {}+ 2k. \end{aligned}$

约去 $x^2$ 后得到 $(k - 5)x = 2k - 6.$ 对于列出的 $k=1,2,3,4,$ 中每一个值，都能得到一个有效的 $x$ ，且它既不是使分母为零的 $2$ ，也不是 $6.$ 当 $k=5,$ 时，方程变为 $0\cdot x=4,$ 无解。

所以正确答案是 E。

Cross multiplying gives $(x - 1)(x - 6) = (x - 2)(x - k),$ which expands to $\begin{aligned} x^2 - 7x + 6 &= x^2 - (2 + k)x \\ &\quad {}+ 2k. \end{aligned}$

Cancelling $x^2$ leaves $(k - 5)x = 2k - 6.$ For each listed value $k=1,2,3,4,$ this gives a valid value of $x$ that is neither excluded denominator value $2$ nor $6.$ For $k=5,$ the equation instead becomes $0\cdot x=4,$ which has no solution.

Thus, the correct answer is E.

2002 AMC 10B 第 12 题

答案：E

难度评级：1370

解答：

其他年份的第 12 题