Solving Simultaneous Equations | 联立方程组的解法

📚 Solving Simultaneous Equations | 联立方程组的解法

Simultaneous equations are a core topic in IGCSE Mathematics. They require you to find values for two or more unknowns that satisfy two or more equations at the same time. This guide explains the main methods: elimination, substitution, and graphical solutions, with worked examples and exam tips.

联立方程组是 IGCSE 数学的核心考点。它要求你求出同时满足两个或更多方程的两个或更多未知数的值。本文讲解主要方法：消元法、代入法、图像法，并提供例题与考试技巧。

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

A pair of simultaneous equations is a set of two or more equations that share the same variables. A solution is a pair of values that satisfies every equation at once. For example, the equations 2x + y = 7 and x − y = 2 must both be true for the same values of x and y.

联立方程组由两个或更多方程组成，它们含有相同的变量。一个解就是一组能同时满足每一个方程的值。例如，方程 2x + y = 7 和 x − y = 2 必须对相同的 x 和 y 值同时成立。

For linear simultaneous equations, there is usually exactly one solution, unless the lines are parallel or identical. The solution can be found algebraically or graphically.

对于线性联立方程组，通常恰好有一个解，除非两条直线平行或重合。解可以通过代数方法或图像法求出。

• Example: 2x + y = 7 and x − y = 2. 示例：2x + y = 7 和 x − y = 2。
• The solution is x = 3, y = 1 because 2(3) + 1 = 7 and 3 − 1 = 2. 解为 x = 3，y = 1，因为 2(3) + 1 = 7 且 3 − 1 = 2。

2. The Elimination Method | 消元法

Elimination is usually the fastest method when both equations are linear and the coefficients of one variable can be matched easily. The idea is to add or subtract the equations to remove one variable.

当两个方程都是线性方程，并且某个变量的系数容易配平时，消元法通常是最快的方法。其思路是通过相加或相减来消去一个变量。

Follow these steps:

按照以下步骤操作：

• Write both equations in the form ax + by = c. 把两个方程写成 ax + by = c 的形式。
• Multiply one or both equations so that the coefficients of one variable are equal. 给一个或两个方程乘以适当的数，使某个变量的系数相等。
• Add or subtract the equations to eliminate that variable. 将方程相加或相减，消去该变量。
• Solve the resulting one-variable equation. 解得到的一元方程。
• Substitute back to find the other variable. 代回原方程求另一个变量。

Worked example:

例题：

2x + y = 7

x − y = 2

Add the two equations to eliminate y:

将两个方程相加，消去 y：

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

Substitute x = 3 into x − y = 2:

将 x = 3 代入 x − y = 2：

3 − y = 2 → y = 1

The solution is x = 3, y = 1.

解为 x = 3，y = 1。

3. The Substitution Method | 代入法

Substitution is useful when one equation

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