Mastering Simultaneous Equations for IGCSE Mathematics | 掌握 IGCSE 数学联立方程

📚 Mastering Simultaneous Equations for IGCSE Mathematics | 掌握 IGCSE 数学联立方程

In IGCSE Mathematics, simultaneous equations appear in both the core and extended tiers. They are closely linked to straight-line graphs, algebraic manipulation, and problem solving. A solid understanding of the elimination and substitution methods will help you tackle many exam questions confidently, especially in Paper 2 and Paper 4.

在 IGCSE 数学中，联立方程在核心课程和拓展课程中都会出现。它们与直线图像、代数运算和问题解决密切相关。扎实掌握消元法和代入法能帮助你自信应对许多考试题目，尤其是在 Paper 2 和 Paper 4 中。

1. What Are Simultaneous Equations? | 什么是联立方程？

A simultaneous equation system is a set of two or more equations that share the same variables. In IGCSE Mathematics, you usually work with two equations in two unknowns, such as 2x + y = 7 and x − y = 2. The solution is a pair of values (x, y) that makes both equations true at the same time. This is why the word ‘simultaneous’ is used.

联立方程组是由两个或更多共享相同变量的方程组成的集合。在 IGCSE 数学中，你通常会处理两个未知数的两个方程，例如 2x + y = 7 和 x − y = 2。解是一对数值 (x, y)，它同时使两个方程都成立。这就是使用 “simultaneous” 一词的原因。

Graphically, each linear equation represents a straight line. The simultaneous solution corresponds to the point where the two lines cross. If the lines intersect at (3, 1), then x = 3 and y = 1 is the unique solution.

从图形上看，每个线性方程表示一条直线。联立方程的解对应两条直线相交的点。如果两条直线在 (3, 1) 处相交，那么 x = 3 和 y = 1 就是唯一解。

2. The Graphical Method | 图像法

To solve a linear system graphically, rearrange each equation into the slope-intercept form y = mx + c. For example, 2x + y = 7 becomes y = −2x + 7, and x − y = 2 becomes y = x − 2. Plot both lines on the same coordinate grid. The point of intersection gives the solution.

要用图像法解线性方程组，先把每个方程改写为斜截式 y = mx + c。例如，2x + y = 7 可写成 y = −2x + 7，x − y = 2 可写成 y = x − 2。在同一坐标系中画出两条直线，交点即为解。

2x + y = 7 → y = −2x + 7
x − y = 2 → y = x − 2

In this example, the line y = −2x + 7 and the line y = x − 2 meet at (3, 1). The graphical method is useful for visual understanding, but it can be less accurate when the solution involves fractions or decimals. For exam accuracy, algebraic methods are usually preferred unless the question asks for a graph.

在这个例子中，直线 y = −2x + 7 和 y = x − 2 相交于 (3, 1)。图像法有助于直观理解，但当解包含分数或小数时可能不够精确。为了考试准确性，通常优先使用代数方法，除非题目要求画图。

3. The Elimination Method | 消元法

The elimination method is often the fastest way to solve linear simultaneous equations. The idea is to add or subtract the equations so that one variable cancels out. Take the system 2x + y = 7 and x − y = 2. If you add the left-hand sides and the right-hand sides, the y terms cancel because y + (−y) = 0.

消元法通常是解线性联立方程最快的方法。它的思路是将两个方程相加或相减，使其中一个变量被消去。以方程组 2x + y = 7 和 x − y = 2 为例。如果把左边和右边分别相加，y 项会抵消，因为 y + (−y) = 0。

(2x + y) + (x − y) = 7 + 2 → 3x = 9 → x = 3

Once x = 3 is found, substitute it back into either original equation. Using x − y = 2 gives 3 − y = 2, so y = 1. The solution is x = 3, y = 1.

一旦求出 x = 3，就把它代回任意一个原方程。使用 x − y = 2，得到 3 − y = 2，所以 y = 1。解为 x = 3，y = 1。

When the coefficients of one variable are not equal, multiply one or both equations by suitable numbers. For example, in 3x + 2y = 12 and x + y = 5, multiply the second equation by 2 to get 2x + 2y = 10. Subtracting this from the first equation gives x = 2, then y =

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