📚 IGCSE Maths G-8 Teacher's Book 687: Straight Line Graphs and Linear Equations | IGCSE 数学 G-8 教师用书 687：直线图像与线性方程
near equations, which appears frequently in IGCSE Mathematics and is often covered in Teacher' c, calculate gradients, plot lines, solve simultaneous equations graphically, and apply these ideas to real-world problems.
本复习指南聚焦直线图像与线性方程这一核心主题，该主题在 IGCSE 数学中经常出现，也常出现在教师用书 G-8 第 687 页附近的题目中。你将学习如何理解 y = mx + c、计算梯度、绘制直线、用图像解联立方程，并将这些方法应用于实际问题。
1. Understanding the Coordinate Plane | 理解坐标平面
ten as an ordered pair (x, y), where x gives the horizontal distance and y gives the vertical distance from the origin.
坐标平面由水平的 x 轴和竖直的 y 轴组成。两轴相交的点是原点 O = (0, 0)。每个点都写成有序数对 (x, y)，其中 x 表示到原点的水平距离，y 表示到原点的竖直距离。
The axes divide the plane into four quadrants. In the first quadrant both x and y are positive; in the third quadrant both are negative. This idea is essential before plotting any straight line.
坐标轴将平面分成四个象限。第一象限中 x 和 y 均为正；第三象限中两者均为负。在绘制任何直线之前，理解这一点非常重要。
2. What is a Linear Equation? | 什么是线性方程？
A linear equation is any equation whose graph is a straight line. The most useful form for IGCSE is the slope-intercept form:
线性方程是指图像为直线的任何方程。对 IGCSE 最有用的形式是斜截式：
y = mx + c
Here m represents the gradient (slope) of the line, and c represents the y-intercept, which is the y-value where the line crosses the y-axis.
这里 m 表示直线的梯度（斜率），c 表示 y 轴截距，即直线与 y 轴相交处的 y 值。
- If m = 2 and c = 1, the equation is y = 2x + 1. | 如果 m = 2，c = 1，则方程为 y = 2x + 1。
- If m = −3 and c = 0, the equation is y = −3x. | 如果 m = −3，c = 0，则方程为 y = −3x。
3. Finding the Gradient | 求梯度
The gradient measures how steep a line is. Given two points (x₁, y₁) and (x₂, y₂), the gradient is calculated by:
梯度衡量直线的倾斜程度。给定两点 (x₁, y₁) 和 (x₂, y₂)，梯度用以下公式计算：
m = (y₂ − y₁) ÷ (x₂ − x₁)
Always subtract the y-coordinates in the same order as the x-coordinates. A positive gradient rises from left to right, while a negative gradient falls from left to right.
计算时 y 坐标相减的顺序一定要与 x 坐标相减的顺序一致。正梯度从左到右上升，负梯度从左到右下降。
A horizontal line has gradient 0, and a vertical line has an undefined gradient because the denominator x₂ − x₁ would be zero.
水平线的梯度为 0，竖直线的梯度无定义，因为分母 x₂ − x₁ 会为零。
4. Interpreting the y-intercept | 理解 y 轴截距
the equation y = 4x − 5, the y-intercept is −5, so the line crosses the y-axis at (0, −5).
y 轴截距 c 是当 x = 0 时 y 的值。在方程 y = 4x − 5 中，y 轴截距为 −5，因此直线与 y 轴相交于点 (0, −5)。
In real-life graphs, c often represents a fixed initial amount, such as a starting fee or base cost before any units are used.
在实际生活图像中，c 常表示固定的初始量，例如开始计费前的起步费或基础成本。
5. Plotting a Straight Line | 绘制直线
rom the y-intercept (0, c), then use the gradient m = rise ÷ run to move to another point.
绘制直线只需要两个点，但第三个点有助于检验。从 y 轴截距 (0, c) 出发，然后利用梯度 m = 上升距离 ÷ 水平距离移动到另一个点。
ve 1 unit right and 2 units up to reach (1, 5). Draw a straight line through these points.
对于 y = 2x + 3，从 (0, 3) 开始。由于 m = 2 = 2/1，向右移动 1 个单位、向上移动 2 个单位到达 (1, 5)。通过这些点画出直线。
Alternatively, create a table of x-values such as −2, −1, 0, 1, 2, calculate the corresponding y-values, and plot the ordered pairs.
也可以列出 x 值表，例如 −2、−1、0、1、2，计算对应的 y 值，然后描出这些有序数对。
6. Parallel and Perpendicular Lines | 平行与垂直直线
ame gradient. For example, y = 3x + 1 and y = 3x − 4 are parallel because both have m = 3.
平行直线具有相同的梯度。例如 y = 3x + 1 和 y = 3x − 4 平行，因为两者的梯度都是 m = 3。
Perpendicular lines have gradients that multiply to give −1. If one line has gradient m, a perpendicular line has gradient −1 ÷ m, often written as −1/m.
垂直直线的梯度相乘等于 −1。如果一条直线的梯度为 m，那么与它垂直的直线梯度为 −1 ÷ m，通常写作 −1/m。
m₁ × m₂ = −1
For instance, if a line has gradient 2, a line perpendicular to it has gradient −1/2.
例如，如果一条直线的梯度为 2，那么与它垂直的直线梯度为 −1/2。
7. Rearranging into y = mx + c | 化成 y = mx + c 形式
px + qy + r = 0. Before identifying m and c, rearrange the equation to make y the subject.
许多 IGCSE 题目给出的方程是 ax + by = c 或 px + qy + r = 0 的形式。在确定 m 和 c 之前，需要先把方程变形为以 y 为主语的形式。
e, 3x + 2y = 6 can be rearranged by subtracting 3x from both sides and then dividing by 2:
例如，3x + 2y = 6 可以通过两边减去 3x，再除以 2 来变形：
2y = −3x + 6 → y = −(3/2)x + 3
Now it is clear that the gradient is −3/2 and the y-intercept is 3.
现在可以清楚地看到梯度为 −3/2，y 轴截距为 3。
8. Finding the Equation from Two Points | 由两点求方程
If you know two points on a line, you can find the equation in three steps. First calculate the gradient m using (y₂ − y₁) ÷ (x₂ − x₁). Then substitute the
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