Mastering Linear Equations: From Balancing to Word Problems | 掌握一次方程：从平衡法到应用题

📚 Mastering Linear Equations: From Balancing to Word Problems | 掌握一次方程：从平衡法到应用题

Linear equations are one of the most important building blocks in KS3 mathematics. They appear when we solve puzzles, calculate unknown quantities, and model real-life situations such as mobile phone tariffs, pocket money savings, or the sides of a rectangle. This article explains the key methods clearly, including balancing, brackets, fractions, and word problems, so that you can work confidently through Cambridge-style questions.

一次方程是 KS3 数学中最重要的基础模块之一。我们在解谜、计算未知量以及模拟实际生活情境（如手机话费、零花钱储蓄或长方形的边长）时都会用到它。本文将清晰地讲解关键方法，包括平衡法、去括号、去分母和应用题，帮助你自信地应对剑桥风格的题目。

1. What Is a Linear Equation? | 什么是一次方程？

A linear equation is an equation in which the unknown appears only to the power 1. For example, x + 3 = 7, 2x − 5 = 11, and 3(x + 2) = 21 are all linear equations. There are no terms like x², x³, or 1/x. The word ‘linear’ indicates that the graph of the equation is a straight line.

一次方程是指未知数只出现一次幂（指数为 1）的方程。例如 x + 3 = 7、2x − 5 = 11 和 3(x + 2) = 21 都是一次方程。方程中不会出现 x²、x³ 或 1/x 这样的项。’一次’ 一词表示方程的图像是一条直线。

• ax + b = c — 一般形式，未知数系数为 a
• a(x + b) = c — 含括号形式
• ax + b = cx + d — 两边都含未知数

When we solve a linear equation, we are looking for the value of the unknown that makes the equation true. This value is called the solution or the root.

解一次方程时，我们要找到使等式成立的未知数的值。这个值叫做方程的解或根。

x + 3 = 7 ⇒ x = 4

Here, x = 4 is the only solution, because 4 + 3 = 7.

这里 x = 4 是唯一解，因为 4 + 3 = 7。

2. The Balance Method | 平衡法

The balance method is the foundation of solving linear equations. Imagine the equation as a pair of scales in perfect balance. Whatever you do to one side, you must do to the other side to keep the scales balanced. This is sometimes called the ‘golden rule’ of algebra.

平衡法是解一次方程的基础。把方程想象成一台完全平衡的天平。无论你对方程的一边做什么，都必须对另一边做同样的操作，才能保持天平平衡。这有时被称为代数的 ‘黄金法则’。

The main operations are adding, subtracting, multiplying, or dividing both sides by the same non-zero number. For example, to solve x + 5 = 12, subtract 5 from both sides. This leaves x on its own.

主要操作是对两边同时加、减、乘或除以同一个非零数。例如，解 x + 5 = 12 时，两边同时减去 5，就可以把 x 单独留在一边。

x + 5 = 12 ⇒ x + 5 − 5 = 12 − 5 ⇒ x = 7

Check by substituting x = 7 back into the original equation: 7 + 5 = 12, which is true.

把 x = 7 代回原方程检验：7 + 5 = 12，等式成立。

3. Solving One-Step Equations | 解一步方程

One-step equations require only one operation to isolate the unknown. They are usually the first type you meet in Year 7

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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