Solving Simultaneous Equations | 解联立方程

📚 Solving Simultaneous Equations | 解联立方程

Simultaneous equations are one of the most frequently tested topics in IGCSE Mathematics. Whether you sit the Core paper or the Extended paper, you need to be able to solve two linear equations with two unknowns, interpret the solution graphically, and apply the same skills to word problems. This revision guide explains each method step by step, highlights common traps, and gives you targeted practice questions.

联立方程是 IGCSE 数学中最高频的考点之一。无论你参加 Core（核心）还是 Extended（扩展）试卷，都需要掌握解含两个未知数的两个线性方程、从图像上解释解，以及运用相同技能解决应用题。本复习指南逐步讲解每一种方法、指出常见陷阱，并为你提供针对性的练习题。

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

A linear equation in two unknowns, such as x + y = 7, has infinitely many solutions by itself. Any point on the line is a solution. When we combine it with a second line, such as x − y = 1, the pair of equations generally has exactly one common solution: the coordinates of the point where the two lines cross.

含两个未知数的线性方程，如 x + y = 7，本身有无数个解，直线上任意一点都是一个解。当我们把它与另一条直线 x − y = 1 组合起来时，这对方程通常只有一组公共解，也就是两条直线交点的坐标。

公共解必须同时满足两个方程：x + y = 7 且 x − y = 1 → x = 4, y = 3

• Algebraic methods: elimination and substitution give exact answers.

  代数方法：消元法和代入法可以得到精确答案。

• Graphical method: the intersection point of two lines gives the solution.

  图像方法：两条直线的交点就是方程组的解。

2. The Elimination Method | 消元法

The elimination method works by removing one variable so that only a single equation in one unknown remains. First, write both equations in the same form, such as ax + by = c. Then make the coefficients of one variable equal in absolute value by multiplying an equation if necessary. If the signs are the same, subtract the equations; if the signs

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