AMC10 2018 A

AMC10 2018 A · Q12

AMC10 2018 A · Q12. It mainly tests Systems of equations, Absolute value.

How many ordered pairs of real numbers \((x, y)\) satisfy the following system of equations?\n\[ \begin{cases} x + 3y = 3 \\ |x| - |y| = 1 \end{cases} \]

多少有序实数对 \((x, y)\) 满足下列方程组？\n\[ \begin{cases} x + 3y = 3 \\ |x| - |y| = 1 \end{cases} \]

(A) 1 1

(B) 2 2

(D) 4 4

(E) 8 8

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): The graph of the system is shown below. The graph of the first equation is a line with \(x\)-intercept \((3,0)\) and \(y\)-intercept \((0,1)\). To draw the graph of the second equation, consider the equation quadrant by quadrant. In the first quadrant \(x>0\) and \(y>0\), and thus the second equation is equivalent to \(|x-y|=1\), which in turn is equivalent to \(y=x\pm 1\). Its graph consists of the rays with endpoints \((0,1)\) and \((1,0)\), as shown. In the second quadrant \(x<0\) and \(y>0\). The corresponding graph is the reflection of the first quadrant graph across the \(y\)-axis. The rest of the graph can be sketched by further reflections of the first-quadrant graph across the coordinate axes, resulting in the figure shown. There are 3 intersection points: \((-3,2)\), \((0,1)\), and \(\left(\frac{3}{2},\frac{1}{2}\right)\), as shown.

答案（C）：该方程组的图像如下所示。第一个方程的图像是一条直线，其 \(x\) 截距为 \((3,0)\)，\(y\) 截距为 \((0,1)\)。为了作出第二个方程的图像，按象限逐一考虑该方程。在第一象限中 \(x>0\) 且 \(y>0\)，因此第二个方程等价于 \(|x-y|=1\)，进而等价于 \(y=x\pm 1\)。其图像由以 \((0,1)\) 和 \((1,0)\) 为端点的射线组成，如图所示。在第二象限中 \(x<0\) 且 \(y>0\)，对应的图像是第一象限图像关于 \(y\) 轴的对称。其余部分可通过将第一象限的图像继续关于坐标轴作对称得到，从而得到图中所示的图形。共有 3 个交点：\((-3,2)\)、\((0,1)\) 和 \(\left(\frac{3}{2},\frac{1}{2}\right)\)，如图所示。

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