Straight Line Graphs and Simultaneous Equations | 直线图像与联立方程

📚 Straight Line Graphs and Simultaneous Equations | 直线图像与联立方程

In KS3 Cambridge Mathematics, straight line graphs and simultaneous equations form a core part of algebra and coordinate geometry. Understanding how to plot linear equations, interpret gradient and y-intercept, and find the intersection of two lines gives you a powerful toolkit for solving problems both on paper and in real-life situations.

在 KS3 剑桥数学中，直线图像与联立方程是代数与坐标几何的核心内容。掌握如何绘制一次函数图像、理解斜率和 y 轴截距，以及找到两条直线的交点，能为你解决书面问题和实际生活中的问题提供强大的工具。

1. The Coordinate Plane and Plotting Points | 坐标平面与描点

A set of coordinates always uses the form (x, y), where x tells you how far to move horizontally from the origin and y tells you how far to move vertically. The point (4, −3) means move 4 units right and 3 units down. The whole plane is divided into four quadrants by the x-axis and y-axis.

坐标始终使用 (x, y) 的形式，其中 x 表示从原点水平移动的距离，y 表示垂直移动的距离。点 (4, −3) 表示向右移动 4 个单位，再向下移动 3 个单位。整个平面被 x 轴和 y 轴分成四个象限。

Plotting several points from a rule such as y = 2x − 1 helps you see that the points lie on a straight line. This is why equations of the form y = mx + c are called linear equations.

根据规则（如 y = 2x − 1）描出几个点，你会发现这些点都落在同一条直线上。这就是形如 y = mx + c 的方程被称为线性方程的原因。

2. Understanding the Equation y = mx + c | 理解直线方程 y = mx + c

In the equation y = mx + c, the letter m represents the gradient (slope) of the line, and c represents the y-intercept. The gradient tells you how steep the line is and whether it goes up or down. The y-intercept is the exact point where the line crosses the y-axis, so it is the value of y when x = 0.

在方程 y = mx + c 中，字母 m 表示直线的斜率（坡度），c 表示 y 轴截距。斜率告诉你直线有多陡，以及它是上升还是下降。y 轴截距是直线与 y 轴相交的确切位置，也就是当 x = 0 时 y 的值。

y = mx + c

If m is positive, the line rises from left to right; if m is negative, it falls. If m = 0, the line is horizontal and has the equation y = c.

如果 m 为正，直线从左到右上升；如果 m 为负，直线下降；如果 m = 0，直线是水平的，其方程为 y = c。

3. Finding Gradient from Two Points | 通过两点求斜率

To calculate the gradient of a line passing through two known points, use the formula below. Always subtract the y-coordinates and x-coordinates in the same order to avoid sign errors.

要计算经过两个已知点的直线的斜率，请使用以下公式。始终以相同的顺序相减 y 坐标和 x 坐标，以避免符号错误。

m = Δy ÷ Δx = (y₂ − y₁) ÷ (x₂ − x₁)

For example, given points A(1, 2) and B(3, 8), identify

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