📚 2. Equations of Straight Lines | 直线方程
In coordinate geometry, a straight line can be described exactly by an equation. Understanding how to find its gradient, write its equation in different forms, and use the relationships between parallel and perpendicular lines is essential for A-Level Mathematics. This article covers the key methods and exam techniques needed for Edexcel coordinate geometry questions.
在坐标几何中，直线可以用方程精确描述。掌握如何求直线的斜率、用不同形式写出直线方程，以及利用平行线和垂直线之间的关系，对于 A-Level 数学至关重要。本文涵盖 Edexcel 坐标几何题目所需的关键方法和应试技巧。
1. Gradient of a Straight Line | 直线的斜率
The gradient measures how steep a line is and the direction it travels. For two points (x₁, y₁) and (x₂, y₂) on a line, the gradient m is the change in y divided by the change in x.
斜率衡量直线的倾斜程度和方向。对于直线上的两点 (x₁, y₁) 和 (x₂, y₂)，斜率 m 等于 y 的变化量除以 x 的变化量。
m = (y₂ − y₁)/(x₂ − x₁)
A positive gradient means the line rises from left to right, a negative gradient means it falls, a zero gradient gives a horizontal line, and a vertical line has an undefined gradient because the change in x is zero.
斜率为正表示直线从左到右上升，斜率为负表示直线下降，斜率为零得到水平线，而竖直线因为 x 的变化量为零，所以斜率无定义。
2. The Equation y = mx + c | 斜截式方程
The most common form of a straight-line equation is y = mx + c, where m is the gradient and c is the y-intercept. The y-intercept is the value of y when x = 0.
直线方程最常见的形式是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。y 轴截距是当 x = 0 时 y 的值。
y = mx + c
For example, y = 3x − 2 has gradient 3 and crosses the y-axis at (0, −2). This form is useful when the gradient and intercept are directly given or can be read from a graph.
例如，y = 3x − 2 的斜率为 3，并在 (0, −2) 处穿过 y 轴。当斜率和截距直接给出或可从图中读出时，这种形式非常有用。
3. Point-Gradient Form | 点斜式
When the gradient m and one point (x₁, y₁) on the line are known, the point-gradient form is the quickest way to write the equation. It is especially useful when the y-intercept is not given.
当已知直线的斜率 m 和直线上一点 (x₁, y₁) 时，点斜式是写出直线方程最快的方法。在不知道 y 轴截距时，这种形式尤其有用。
y − y₁ = m(x − x₁)
This can always be rearranged into y = mx + c. For instance, a line through (2, 5) with gradient 4 has equation y − 5 = 4(x − 2), which simplifies to y = 4x − 3.
这个形式总能整理成 y = mx + c。例如，经过点 (2, 5) 且斜率为 4 的直线方程为 y − 5 = 4(x − 2)，化简后得到 y = 4x − 3。
4. Finding the Equation from Two Points | 由两点求方程
If two points on a line are given, first calculate the gradient using m = (y₂ − y₁)/(x₂ − x₁). Then substitute one of the points into the point-gradient form.
如果给出直线上的两个点，首先用 m = (y₂ − y₁)/(x₂ − x₁) 计算斜率，然后将其中一个点代入点斜式。
m = (y₂ − y₁)/(x₂ − x₁), then y − y₁ = m(x − x₁)
For example, to find the line through (1, 4) and (3, 10), first calculate m = (10 − 4)/(3 − 1) = 3. Using (1, 4), the equation is y − 4 = 3(x − 1), which simplifies to y = 3x + 1. Always check that both points satisfy the final equation.
例如，求经过 (1, 4) 和 (3, 10) 两点的直线，先计算 m = (10 − 4)/(3 − 1) = 3。代入 (1, 4) 得到 y − 4 = 3(x − 1)，化简为 y = 3x + 1。务必检查两个点都满足最终方程。
5. General Form ax + by + c = 0 | 一般式
A straight line can also be written in the general form ax + by + c = 0, where a, b and c are integers and a is usually positive. Exam questions often ask for the final answer in this form.
直线也可以写成一般式 ax + by + c = 0，其中 a、b 和 c 为整数，且通常要求 a 为正数。考试题经常要求最终答案写成这种形式。
ax + by + c = 0
To convert from y = mx + c, rearrange all terms to one side. For example, y = (2/3)x − 1 becomes 2x − 3y − 3 = 0 after multiplying through by 3 to remove the fraction.
从 y = mx + c 转换时，把所有项移到一边。例如，y = (2/3)x − 1 两边乘以 3 去掉分数后，可化为 2x − 3y − 3 = 0。
6. Parallel Lines | 平行线
Parallel lines have the same gradient. If line L₁ has gradient m₁ and line L₂ has gradient m₂, then L₁ is parallel to L₂ if and only if m₁ = m₂.
平行线具有相同的斜率。如果直线 L₁ 的斜率为 m₁，直线 L₂ 的斜率为 m₂，那么 L₁ 平行于 L₂ 当且仅当 m₁ = m₂。
m₁ = m₂
To find a line parallel to y = 5x + 2 and passing through (1, 4), keep the gradient 5 and use the point-gradient form: y − 4 = 5(x − 1), which gives y = 5x − 1.
要求一条平行于 y = 5x + 2 且经过点 (1, 4) 的直线，保持斜率为 5 并使用点斜式：y − 4 = 5(x − 1)，得到 y = 5x − 1。
7. Perpendicular Lines | 垂直线
Perpendicular lines have gradients whose product is −1. If m₁ and m₂ are the gradients of two perpendicular lines, then m₁ × m₂ = −1, which is equivalent to m₂ = −1/m₁.
垂直线的斜率乘积为 −1。如果 m₁ 和 m₂ 是两条垂直线的斜率，那么 m₁ × m₂ = −1，也就是 m₂ = −1/m₁。
m₁ × m₂ = −1
For example, a line perpendicular to y = 2x + 3 has gradient −1/2. Note that vertical and horizontal lines are also perpendicular: a vertical line has undefined gradient, while a horizontal line has gradient 0.
例如，垂直于 y = 2x + 3 的直线斜率为 −1/2。注意竖直线和水平线也相互垂直：竖直线的斜率无定义，水平线的斜率为 0。
8. Midpoint and Distance | 中点与距离
The midpoint of two points (x₁, y₁) and (x₂, y₂) is found by averaging the x-coordinates and the y-coordinates separately. This is often used to locate the centre of a line segment or to find a perpendicular bisector.
两点 (x₁, y₁) 和 (x₂, y₂) 的中点通过分别对 x 坐标和 y 坐标求平均得到。这常用于确定线段的中点或求垂直平分线。
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The distance between the two points comes from Pythagoras’ theorem in the coordinate plane. It gives the exact length of the line segment joining them.
两点之间的距离来自坐标平面中的勾股定理。它给出了连接两点的线段的精确长度。
d = √((x₂ − x₁)² + (y₂ − y₁)²)
9. Intersection of Two Lines | 两直线的交点
To find where two lines meet, solve their equations simultaneously. If the equations are given as y = m₁x + c₁ and y = m₂x + c₂, set them equal to each other and solve for x, then substitute back to find y.
求两条直线的交点时，需要联立解它们的方程。如果方程分别为 y = m₁x + c₁ 和 y = m₂x + c₂，令两者相等解出 x，然后代回求出 y。
m₁x + c₁ = m₂x + c₂
If the lines are parallel and distinct, there is no intersection. If the two equations represent the same line, there are infinitely many intersection points. Otherwise, the simultaneous equations give a unique solution.
如果两条直线平行且不重合，则没有交点。如果两个方程表示同一条直线，则有无穷多个交点。否则，联立方程组有唯一解。
10. Modelling with Straight
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