📚 IGCSE Maths Linear Graphs and Simultaneous Equations | IGCSE 数学线性图像与联立方程
m a central part of the IGCSE Mathematics syllabus. This article, based on the G-7 Teacher' nes, interpret y = mx + c, and apply algebraic and graphical techniques to find solutions.
线性图像与联立方程是 IGCSE 数学课程的核心内容。本文依据 G-7 教师用书第 200 页的主题，解释斜率、截距、直线方程以及求解两个线性方程的常用方法。你将学习如何绘制直线图像、理解 y = mx + c 的含义，并运用代数法和图像法求出联立方程的解。
1. Coordinates and the Cartesian Plane | 坐标与笛卡尔平面
In IGCSE coordinate geometry, every point is described by an ordered pair (x, y). The first number is the x-coordinate measured horizontally from the origin, and the second number is the y-coordinate measured vertically. The origin is the point (0, 0), where the x-axis and y-axis cross.
在 IGCSE 坐标几何中，每个点用一个有序数对 (x, y) 表示。第一个数是 x 坐标，从原点沿水平方向测量；第二个数是 y 坐标，从原点沿竖直方向测量。原点是 (0, 0)，它是 x 轴与 y 轴的交点。
When plotting points, start at the origin, move right for positive x and left for negative x, then move up for positive y and down for negative y. For example, the point (3, -2) is 3 units to the right and 2 units down from the origin.
描点时，从原点出发，x 为正向右移动，x 为负向左移动；然后 y 为正向上移动，y 为负向下移动。例如，点 (3, -2) 位于原点右侧 3 个单位、下方 2 个单位处。
The plane is divided into four quadrants. In the first quadrant both x and y are positive. In the second quadrant x is negative and y is positive. In the third quadrant both are negative, and in the fourth quadrant x is positive while y is negative.
坐标平面分为四个象限。第一象限中 x 和 y 都为正；第二象限中 x 为负、y 为正；第三象限中两者都为负；第四象限中 x 为正、y 为负。
2. Plotting Linear Graphs from a Table of Values | 通过数值表绘制线性图像
e plotted by choosing x-values, substituting them into the equation, and finding the matching y-values. A table of values helps organise the calculations before plotting.
对于 y = 2x + 1 这样的线性方程，可以通过选取 x 值、代入方程并求出相应的 y 值来绘制图像。在描点之前，使用数值表有助于整理计算过程。
- Choose at least three x-values, for example x = -2, 0, 2.
- Work out y = 2(-2) + 1 = -3, y = 2(0) + 1 = 1, y = 2(2) + 1 = 5.
- Plot the points (-2, -3), (0, 1), (2, 5) and join them with a straight line.
至少选取三个 x 值，例如 x = -2、0、2；分别计算 y = 2(-2) + 1 = -3、y = 2(0) + 1 = 1、y = 2(2) + 1 = 5；然后描出 (-2, -3)、(0, 1)、(2, 5)，并用直线连接。
Choosing x-values that are spread out helps to reduce drawing errors. If one plotted point does not lie on the same straight line as the others, check the substitution and the point position carefully.
选择分布较开的 x 值有助于减少绘图误差。如果某个描出的点不与其他点在同一条直线上，要仔细检查代入计算和点的位置。
3. Gradient of a Straight Line | 直线的斜率
The gradient measures how steep a line is. It is calculated by dividing the vertical change by the horizontal change between any two points on the line. The gradient is usually represented by the letter m.
斜率用来衡量一条直线的倾斜程度。它等于直线上任意两点之间的竖直变化量除以水平变化量。斜率通常用字母 m 表示。
m = (y₂ - y₁) / (x₂ - x₁)
ve gradient slopes upwards from left to right, while a negative gradient slopes downwards.
例如，一条直线经过 (1, 2) 和 (3, 8)，其斜率 m = (8 - 2) / (3 - 1) = 6 / 2 = 3。正斜率的直线从左到右向上倾斜，负斜率的直线从左到右向下倾斜。
A horizontal line has gradient 0 because the vertical change is zero. A vertical line has undefined gradient because the horizontal change is zero and division by zero is not defined in mathematics.
水平线的斜率为 0，因为竖直变化量为零。竖直线的斜率未定义，因为水平变化量为零，而在数学中除以零没有定义。
4. y = mx + c and Intercepts | y = mx + c 与截距
nd c is the y-intercept. The y-intercept is the point where the line crosses the y-axis, and it has coordinates (0, c).
直线的一般方程是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。y 轴截距是直线与 y 轴的交点，其坐标为 (0, c)。
intercept is found by setting y = 0 and solving for x, which gives 0 = 3x - 2, so x = 2/3.
对于 y = 3x - 2，斜率为 3，直线与 y 轴交于 (0, -2)。令 y = 0 并解方程，可以得到 x 轴截距：0 = 3x - 2，所以 x = 2/3。
| Equation 方程 | Gradient 斜率 | y-intercept y 截距 |
| y = 2x + 5 | 2 | (0, 5) |
| y = -x + 4 | -1 | (0, 4) |
ed to y = mx + c by making y the subject. For example, 2x + 3y = 6 becomes y = -2/3 x + 2.
如果方程以 Ax + By = C 的形式给出，可以通过把 y 写成主项将其转化为 y = mx + c。例如，2x + 3y = 6 可以写成 y = -2/3 x + 2。
5. Finding the Equation from a Graph | 根据图像求方程
ind the gradient using m = (y₂ - y₁) / (x₂ - x₁), then substitute m and c into y = mx + c.
如果直线图像显示 y 轴截距为 (0, c)，并且知道直线上另一个清晰的点，首先用 m = (y₂ - y₁) / (x₂ - x₁) 求斜率，然后将 m 和 c 代入 y = mx + c。
(-1)) / (4 - 0) = 8 / 4 = 2, and the y-intercept is c = -1, so the equation is y = 2x - 1.
例如，一条直线经过 (0, -1) 和 (4, 7)。斜率为 (7 - (-1)) / (4 - 0) = 8 / 4 = 2，y 轴截距为 c = -1，因此方程为 y = 2x - 1。
using two points. First find m, then substitute one point into y = mx + c and solve for c.
如果 y 轴截距没有清晰地显示出来，仍然可以利用两个点求方程。先求出 m，再将其中一个点代入 y = mx + c 并解出 c。
6. Parallel and Perpendicular Lines | 平行线与垂直线
Parallel lines have the same gradient. In IGCSE exam questions, you may be asked to find the equation of a line parallel to a given line through a given point.
平行线具有相同的斜率。在 IGCSE 考试中，可能会要求你求过给定点且
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