Coordinate Geometry: Equations of Straight Lines | 坐标几何：直线方程

📚 Coordinate Geometry: Equations of Straight Lines | 坐标几何：直线方程

Coordinate geometry, also known as analytical geometry, connects algebra and geometry by describing shapes using equations. In this section of the workbook, we focus on straight lines on the Cartesian plane: how to measure their steepness, how to write their equations, and how to analyse relationships between them.

坐标几何，又称解析几何，通过方程将代数与几何联系起来。在本节练习册内容中，我们重点研究笛卡尔平面上的直线：如何度量它们的倾斜程度、如何写出它们的方程，以及如何分析直线之间的关系。

1. The Cartesian Plane | 笛卡尔平面

The Cartesian plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They intersect at the origin O (0, 0).

笛卡尔平面由两条互相垂直的数轴组成：水平的 x 轴和竖直的 y 轴，它们相交于原点 O (0, 0)。

Every point on the plane is described by an ordered pair (x, y). The first number gives the horizontal position measured from the y-axis; the second gives the vertical position measured from the x-axis.

平面上的每个点都用有序数对 (x, y) 来描述，第一个数表示从 y 轴量起的水平位置，第二个数表示从 x 轴量起的竖直位置。

• The x-coordinate is positive to the right of the y-axis and negative to the left.
• x 坐标在 y 轴右侧为正，左侧为负。
• The y-coordinate is positive above the x-axis and negative below it.
• y 坐标在 x 轴上方为正，下方为负。
• The axes divide the plane into four quadrants, numbered I, II, III and IV anticlockwise.
• 坐标轴将平面划分为四个象限，按逆时针方向依次编号为 I、II、III、IV。

2. Gradient of a Line | 直线的斜率

The gradient (or slope) of a straight line measures how steeply it rises or falls as we move from left to right. It is defined as the change in y divided by the change in x.

直线的斜率用来度量直线从左向右移动时的陡峭程度，定义为 y 的变化量除以 x 的变化量。

For two points A(x₁, y₁) and B(x₂, y₂) on the same line, the gradient m is calculated using the formula:

对于同一直线上的两点 A(x₁, y₁) 和 B(x₂, y₂)，斜率 m 通过如下公式计算：

m = (y₂ − y₁) ÷ (x₂ − x₁)

Note that the order of subtraction must be consistent in the numerator and the denominator, but it does not matter which point you label as (x₁, y₁) as long as you keep the same orientation for both coordinates.

请注意：分子和分母中的减法顺序必须保持一致；先标哪个点为 (x₁, y₁) 并不重要，只要两个坐标保持同一方向即可。

• A positive gradient means the line slopes upward from left to right.
• 斜率为正意味着直线从左向右上升。
• A negative gradient means the line slopes downward from left to right.
• 斜率为负意味着直线从左向右下降。
• A horizontal line has gradient 0.
• 水平线的斜率为 0。
• A vertical line has an undefined gradient (division by zero).
• 竖直线的斜率不存在（分母为零）。

3. The Equation y = mx + c | 直线方程 y = mx + c

The most important form of the equation of a straight line is y = mx + c, where m represents the gradient and c represents the y-intercept, that is, the point where the line crosses the y-axis.

直线方程最重要的形式是 y = mx + c，其中 m 表示斜率，c 表示 y 轴截距，即直线与 y 轴交点的纵坐标。

For example, in the equation y = 2x + 3, the gradient is 2 and the y-intercept is 3. This means that for every unit you move to the right, the line rises by 2 units, and it passes through the point (0, 3).

例如，在方程 y = 2x + 3 中，斜率为 2，y 截距为 3。这意味着你每向右移动 1 个单位，直线就上升 2 个单位，并且该直线经过点 (0, 3)。

Any line that is not vertical can be written in this form. Vertical lines are written as x = k, where k is a constant.

任何非竖直的直线都可以写成这种形式。竖直线写作 x = k，其中 k 为常数。

4. Finding the Equation from a Point and the Gradient | 由一点和斜率求直线方程

Suppose you are given the gradient m of a line and the coordinates of one point (x₁, y₁) on the line. You can substitute these values into y = mx + c to find c, and then write the full equation.

假设已知一条直线的斜率 m 以及直线上一点 (x₁, y₁) 的坐标，你可以将这些值代入 y = mx + c 求出 c，然后写出完整的直线方程。

Worked example. A line has gradient 3 and passes through the point (2, 7). Find its equation.

例题。一条直线的斜率为 3，且经过点 (2, 7)，求该直线的方程。

Substitute m = 3, x = 2 and y = 7 into y = mx + c:

将 m = 3，x = 2，y =

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