Simultaneous Equations | 掌握联立方程

📚 Simultaneous Equations | 掌握联立方程

Simultaneous equations are a fundamental topic in IGCSE Mathematics. They appear in almost every exam paper and form the foundation for many advanced topics such as coordinate geometry, vectors, and calculus. In this article, we will break down the three core methods — elimination, substitution, and graphical solution — along with tricky cases involving one linear and one quadratic equation. You will also learn how to translate word problems into equations and avoid the most common pitfalls.

联立方程是 IGCSE 数学中的核心考点，几乎每年必考。它不仅是独立的知识点，更是坐标几何、向量和微积分等高级内容的基础。在本文中，我们将系统讲解三大解法——消元法、代入法和图像法，并深入分析一次方程与二次方程联立的难点。你还将学会如何将文字题转化为方程组，以及如何避开最常见的失分陷阱。

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

Simultaneous equations are a set of two or more equations that share the same unknown variables. In IGCSE Mathematics, you are usually given two equations with two unknowns, typically x and y. The goal is to find the values of x and y that satisfy both equations at the same time.

联立方程是指包含相同未知数的两个或两个以上的方程。在 IGCSE 数学考试中，通常给出两个含有两个未知数（一般为 x 和 y）的方程，目标是找到同时满足这两个方程的一组 x 和 y 的值。

A solution to a pair of simultaneous equations is an ordered pair (x, y) that makes both equations true. For example, x = 2, y = 5 is a solution of the pair y = 3x − 1 and y = x + 3, because both equations give y = 5 when x = 2.

联立方程的解是一个有序数对 (x, y)，它能使两个方程同时成立。例如，x = 2，y = 5 就是联立方程 y = 3x − 1 与 y = x + 3 的一组解，因为当 x = 2 时，两个方程都给出 y = 5。

You should already know that a linear equation in two variables represents a straight line on a graph. Therefore, solving a pair of linear simultaneous equations is equivalent to finding the point where the two lines intersect.

你应该已经知道，含有两个变量的线性方程在图像上表示一条直线。因此，解一组线性联立方程，本质上就是求两条直线的交点坐标。

2. The Elimination Method | 消元法

The elimination method is usually the quickest way to solve linear simultaneous equations. The idea is to eliminate (remove) one variable by adding or subtracting the equations, so that only one variable remains, which can then be solved normally.

消元法通常是解线性联立方程最快的方法。其核心思路是：通过两个方程相加或相减，消去（去掉）其中一个未知数，使方程中只含有一个未知数，再按常规方法求解。

Step 1: Make sure the coefficients of one variable are the same (or opposites) in both equations.

第一步：确保两个方程中某一个未知数的系数相同（或互为相反数）。

Step 2: Add or subtract the equations to eliminate that variable.

第二步：将两个方程相加或相减，以消去该未知数。

Step 3: Solve the resulting one-variable equation.

第三步：解这个一元一次方程，得到一个未知数的值。

Step 4: Substitute the value back into one of the original equations to find the other variable.

第四步：将求出的值代回任意一个原方程中，求出另一个未知数。

Example 1. Solve: 3x + y = 10 (1), x + y = 4 (2)

例 1：解方程组：3x + y = 10 (1)，x + y = 4 (2)

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