Question

正整数 $a$ 和 $b$ 使得直线 $y=ax+5$ 与 $y=3x+b$ 在 $x$ 轴上交于同一点。所有可能交点的 $x$ 坐标之和是多少？

Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$ -axis at the same point. What is the sum of all possible $x$ -coordinates of these points of intersection?

$-20$

$-18$

$-15$

$-12$

$-8$

答案：E

知识点：一次方程因数系统列举

难度评级：1600

解答：

两条直线与 $x$ 轴相交时 $y = 0$ ，所以且解得和 $0 = ax + 5$ $0 = 3x + b,$ $x = -\dfrac{5}{a}$ $x = -\dfrac{b}{3}.$

令二者相等，得到 $\dfrac{5}{a} = \dfrac{b}{3}$ $ab = 15.$

因为 $a$ 和 $b$ 都是正整数，所以 $(a, b)$ 只能是 $(1, 15),$ $(3, 5),$ $(5, 3),$ $(15, 1).$

对应的 $x$ 坐标为它们的和为 $-8$ 。 $x = -5, -\dfrac{5}{3}, -1, -\dfrac{1}{3}.$

所以正确答案是 E。

Note that the lines intersect the $x$ -axis when $y = 0.$ This gives us $0 = ax + 5$ and $0 = 3x + b,$ which when solved gives us $x = -\dfrac{5}{a}$ and $x = -\dfrac{b}{3}.$

Setting these equal to each other, we have $\dfrac{5}{a} = \dfrac{b}{3}$ $ab = 15.$

We know that $a$ and $b$ are positive, which means that the only pairs of values $(a, b)$ that satisfy the above equation are $(1, 15),$ $(3, 5),$ $(5, 3),$ $(15, 1).$

Plugging these values back into the equations gives us $x$ -values of $x = -5, -\dfrac{5}{3}, -1, -\dfrac{1}{3}.$ The sum of all these values is $-8.$

Thus, E is the correct answer.

2014 AMC 10A 第 21 题

答案：E

难度评级：1600

解答：

其他年份的第 21 题