Equations of Lines | 直线方程

📚 Equations of Lines | 直线方程

Lines are the simplest and most important objects in coordinate geometry. Understanding how to represent, manipulate, and interpret straight lines forms a foundational skill for A-Level Mathematics, particularly in the AQA specification.

直线是坐标几何中最简单也最重要的对象。理解如何表示、运算和解释直线，是 A-Level 数学（尤其是 AQA 大纲）中的基础技能。

1. Gradient and y-Intercept Form | 斜率与 y 截距式

The most familiar form of a straight line is \(y = mx + c\). Actually, we must not use LaTeX. So write: The most familiar form of a straight line is y = mx + c. Here \(m\) is the gradient and \(c\) is the y-intercept, the point where the line crosses the y-axis.

最熟悉的直线方程形式是 y = mx + c，其中 m 是斜率（gradient），c 是 y 截距，即直线与 y 轴相交处的纵坐标。

The gradient measures how steep the line is. It is defined as the change in y divided by the change in x between any two points on the line.

斜率衡量直线的倾斜程度。它定义为直线上任意两点之间 y 的变化量除以 x 的变化量。

m = (y₂ − y₁) / (x₂ − x₁)

If m is positive, the line slopes upwards from left to right. If m is negative, the line slopes downwards. If m = 0, the line is horizontal.

如果 m 为正，直线从左到右向上倾斜；如果 m 为负，直线向下倾斜；如果 m = 0，则直线是水平的。

2. General Form ax + by + c = 0 | 一般式 ax + by + c = 0

Another common representation is the general linear form ax + by + c = 0, where a, b, and c are constants and not both a and b are zero.

另一种常见的表示是线性一般式 ax + by + c = 0，其中 a、b、c 是常数，且 a 和 b 不能同时为零。

This form is useful because it can represent any straight line, including vertical lines, which cannot be written in the form y = mx + c. For example, a vertical line x = 3 can be written as 1·x + 0·y − 3 = 0.

这种形式的优点是能表示任意直线，包括垂直直线，而垂直直线无法写成 y = mx + c。例如，竖直线 x = 3 可以写成 1·x + 0·y − 3 = 0。

To convert from the general form to the gradient-intercept form, rearrange to make y the subject:

要从一般式转换为斜率截距式，只需把 y 作为未知数重新排列：

ax + by + c = 0 → y = −(a/b)x − (c/b)

Hence the gradient is −a/b and the y-intercept is −c/b (provided b ≠ 0).

因此斜率为 −a/b，y 截距为 −c/b（当 b ≠ 0 时）。

3. Point-Slope Form | 点斜式

Given a point (x₁, y₁) on a line and the gradient m, the equation of the line can be written as:

已知直线上一点 (x₁, y₁) 和斜率 m，该直线的方程可以写成：

y − y₁ = m(x − x₁)

This is called the point-slope form. It is especially useful when you are given a point and a gradient, or when you derive the equation from two points.

这称为点斜式。当你已知一点和斜率，或需要由两点推导方程时，这个形式尤其方便。

To obtain the gradient-intercept form, expand and simplify: y = mx − mx₁ + y₁, so the y-intercept is y₁ − mx₁.

要得到斜率截距式，只需展开并化简：y = mx − mx₁ + y₁，因此 y 截距为 y₁ − mx₁。

Example: Find the equation of the line through (2, 3) with gradient 4. Using the point-slope form: y − 3 = 4(x − 2), so y = 4x − 5.

例：求经过点 (2, 3) 且斜率为 4 的直线方程。使用点斜式：y − 3 = 4(x − 2)，即 y = 4x − 5。

4. Finding the Gradient from Two Points | 由两点求斜率

If two points (x₁, y₁) and (x₂, y₂) are known, the gradient of the line through them is:

若已知两点 (x₁, y₁) 和 (x₂, y₂)，通过这两点的直线的斜率为：

m = (y₂ − y₁) / (x₂ − x₁)

It is important to subtract the coordinates in the same order in the numerator and denominator. That is, both should be “second point minus first point”.

请注意，分子和分母中的坐标相减顺序必须一致，即都应采用“第二点减第一点”。

Once the gradient is known, substitute either of the two points into the point-slope form to find the full equation of the line.

求出斜率后，将两个点中的任意一个代入点斜式，即可得到直线的完整方程。

Worked example: Find the equation of the line through A(1, 2) and B(4, 8). Gradient m = (8 − 2)/(4 − 1) = 6/3 = 2. Using point A: y − 2 = 2(x − 1) → y = 2x.

例题：求经过 A(1, 2) 和 B(4, 8) 的直线方程。斜率 m = (8 − 2)/(4 − 1) = 6/3 = 2。使用点 A：y − 2 = 2(x − 1)，即 y = 2x。

5. Parallel and Perpendicular Lines | 平行与垂直直线

Two lines are parallel if and only if they have the same gradient. In terms of the general form, lines ax + by + c₁ = 0 and ax + by + c₂ = 0 are parallel because both have gradient −a/b.

两条直线平行当且仅当它们的斜率相等。用一般式表示，直线 ax + by + c₁ = 0 和 ax + by + c₂ = 0 平行，因为它们的斜率都为 −a/b。

Two lines with gradients m₁ and m₂ are perpendicular if and only if m₁ × m₂ = −1. This means m₂ = −1/m₁ (provided neither is zero).

两条斜率分别为 m₁ 和 m₂ 的直线垂直，当且仅当 m₁ × m₂ = −1。也就是说 m₂ = −1/m₁（前提是两者都不为零）。

For example, if one line has gradient 2, a perpendicular line has gradient −½. If one line is horizontal (m = 0), any perpendicular line is vertical (infinite gradient).

例如，若一条直线斜率为 2，则其垂线斜率为 −½。若一条直线是水平的（m = 0），那么任何垂线都是竖直的（斜率为无穷大）。

Sometimes vertical lines require special care: two vertical lines are parallel, and a vertical line is perpendicular to every horizontal line.

竖直直线需要特别注意：两条竖直线互相平行，且竖直线与每条水平线都垂直。

6. Midpoint and Distance Between Points | 中点与两点间距离

Although not strictly an equation of a line, the midpoint and distance formulas are frequently used together with line equations.

虽然中点与距离公式本身不是直线方程，但它们在求解直线问题时经常与直线方程一起使用。

The midpoint M of two points (x₁, y₁) and (x₂, y₂) is:

两点 (x₁, y₁) 和 (x₂, y₂) 的中点 M 为：

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

The distance d between the two points is found using Pythagoras:

两点间的距离 d 由勾股定理求出：

d = √((x₂ − x₁)² + (y₂ − y₁)²)

These formulas are essential when solving problems about the geometry of lines, such as finding the perpendicular bisector of a segment.

这两个公式在求解关于直线几何的问题时至关重要，例如求一条线段的垂直平分线。

7. Intersection of Two Lines | 两直线的交点

The intersection point of two non-parallel lines is the unique point that satisfies both equations simultaneously. To find it, solve the two equations as a pair of simultaneous linear equations.

两条不平行直线的交点是同时满足两个方程的唯一点。要找到交点，需要将两个方程作为联立线性方程组求解。

There are three possibilities for two lines:

两条直线的位置关系有三种可能：

Intersecting: exactly one common point (gradients differ).

相交：恰有一个公共点（斜率不同）。
Parallel: no common points (gradients equal).

平行：没有公共点（斜率相同）。
Coincident: infinitely many common points (same line).

重合：有无穷多个公共点（两条直线相同）。

For example, to solve y = 2x + 1 and y = −x + 4, set 2x + 1 = −x + 4 → 3x = 3 → x = 1. Then y = 3. The intersection is (1, 3).

例如，解 y = 2x + 1 和 y = −x + 4，令 2x + 1 = −x + 4，得 3x = 3，x = 1，则 y = 3。交点为 (1, 3)。

8. Sketching and Interpreting Lines | 直线作图与解读

To sketch a line efficiently, find two key points: the y-intercept (where x = 0) and the x-intercept (where y = 0). Plotting these two points and joining them gives the entire line.

为了高效地画一条直线，找出两个关键点：y 截距（x = 0 处）和 x 截距（y = 0 处）。画出这两点并连接即可得到整条直线。

For example, for y = 3x − 6, the y-intercept is (0, −6). Setting y = 0 gives 0 = 3x − 6 → x = 2, so the x-intercept is (2, 0).

例如，对于 y = 3x − 6，y 截距为 (0, −6)。令 y = 0 得 0 = 3x − 6，即 x = 2，所以 x 截距为 (2, 0)。

When interpreting a line from a graph, read the gradient by measuring the vertical change over the horizontal change using two accurately plotted points. The y-intercept is read directly from the y-axis.

从图中解读直线时，选取两个精确描出的点，用纵向变化量除以横向变化量来读出斜率。y 截距直接从 y 轴上读取。

Be careful with scales: if the axes have different scales, the visual steepness does not equal the mathematical gradient.

要注意刻度：如果两个坐标轴的刻度不同，视觉上的陡峭程度并不等于数学上的斜率值。

9. Worked Exam-Style Example | 考试风格例题

A line passes through the points P(2, 5) and Q(6, 1). Find: (a) the gradient of PQ, (b) the equation of PQ, (c) the equation of the perpendicular bisector of PQ.

一条直线经过点 P(2, 5) 和 Q(6, 1)。求：(a) PQ 的斜率；(b) PQ 的方程；(c) PQ 的垂直平分线的方程。

(a) Gradient m = (1 − 5)/(6 − 2) = (−4)/4 = −1.

(a) 斜率 m = (1 − 5)/(6 − 2) = (−4)/4 = −1。

(b) Using point P: y − 5 = −1(x − 2) → y − 5 = −x + 2 → y = −x + 7.

(b) 使用点 P：y − 5 = −1(x − 2)，即 y − 5 = −x + 2，所以 y = −x + 7。

(c) The midpoint of PQ is ((2+6)/2, (5+1)/2) = (4, 3). The gradient of PQ is −1, so the perpendicular gradient is 1. Therefore the perpendicular bisector has equation y − 3 = 1(x − 4), i.e. y = x − 1.

10. Common Pitfalls and Revision Tips | 常见错误与复习提示

One common mistake is mixing up the order of coordinates when calculating gradient. Always use the same order in the numerator and denominator.

一个常见错误是计算斜率时混淆坐标相减的顺序。分子和分母中必须使用相同的顺序。

Another error is forgetting that vertical lines cannot be expressed in the form y = mx + c. Use the general form or x = constant instead.

另一个错误是忘记竖直直线不能表示为 y = mx + c。此时应使用一般式或 x = 常数。

When checking if two lines are perpendicular, remember that m₁ × m₂ = −1, not m₁ = m₂. The latter condition is for parallel lines.

检查两条直线是否垂直时，要记住条件为 m₁ × m₂ = −1，而不是 m₁ = m₂。后者是平行条件。

Finally, always simplify the equation of a line fully. If you end up with fractions, leave them as fractions or convert to a consistent form such as ax + by + c = 0.

最后，务必把直线方程化到最简。若出现分数，可保留分数或统一转换为 ax + by + c = 0 的形式。

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Equations of Lines | 直线方程