📚 Mastering Simultaneous Equations | 精通联立方程
Simultaneous equations are a core topic in the IGCSE Mathematics syllabus. Whenever two unknown quantities are connected by two independent conditions, we can set up two equations and solve them together. The solution is an ordered pair of values, usually written as (x, y), that makes both equations true at the same time. This topic combines algebraic manipulation, graph interpretation, and real-world modelling, so it appears frequently in Paper 2 and Paper 4.
联立方程是 IGCSE 数学大纲中的核心专题。当两个未知量由两个独立条件联系在一起时，我们可以建立两个方程并一起求解。解是一组有序数对，通常写作 (x, y)，它能同时使两个方程成立。这一专题融合了代数运算、图像解读与实际应用建模，因此在 Paper 2 和 Paper 4 中经常出现。
1. What Are Simultaneous Equations? | 什么是联立方程？
create a system. The two equations represent two straight lines on a graph, and the point where the lines intersect gives the unique solution.
一个含有两个变量的线性方程，如 x + y = 10，有无数个解，因为任意给定一个 x 值都能求出对应的 y 值。当我们再加入一个方程，如 x − y = 2，就形成了一个方程组。这两个方程在图像上分别表示一条直线，两条直线的交点就是唯一解。
There are three standard methods: elimination, substitution, and the graphical method.
三种标准解法：消元法、代入法和图像法。
You must always check your solution in both original equations.
必须始终把解代回两个原方程进行检验。
2. The Elimination Method | 消元法
The elimination method works by adding or subtracting the equations so that one variable cancels out. This requires the coefficients of one variable to be the same, or to have the same absolute value.
消元法的原理是通过两个方程相加或相减，使其中一个变量被抵消。这要求某个变量的系数相同，或绝对值相同。
Consider the following pair:
考虑以下方程组：
3x + y = 11 … (1)
2x − y = 4 … (2)
. Substituting x = 3 into equation (1) gives 9 + y = 11, so y = 2. The solution is (3, 2).
由于 y 项的符号相反，我们把两个方程相加，得到 5x = 15，所以 x = 3。把 x = 3 代入方程 (1) 得 9 + y = 11，所以 y = 2。解为 (3, 2)。
When the coefficients do not match, you must multiply one or both equations before eliminating.
当系数不一致时，必须先对一个或两个方程进行倍数变换，然后再消元。
2x + 3y = 8 … (1)
x − 2y = −3 … (2)
. Substituting back gives x = 1. Always state the final answer clearly as the pair (1, 2).
将方程 (2) 乘以 2，得 2x − 4y = −6。用这个新方程减去方程 (1)：(2x − 4y) − (2x + 3y) = −6 − 8，即 −7y = −14，得 y = 2。代回后得 x = 1。最终答案应清晰写成数对 (1
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