📚 Solving Two-Step Linear Equations: A Complete KS3 Guide | 解两步线性方程：KS3 完整指南
In Cambridge KS3 mathematics, solving equations is a core algebra skill and one of the strongest predictors of later success in IGCSE and A Level. A two-step linear equation is an equation in which exactly two inverse operations have been applied to the unknown. This article breaks the whole process into clear steps, with the balance rule, worked examples, common errors, word problems and practice questions with full answers.
在剑桥 KS3 数学中，解方程是代数核心技能，也是日后 IGCSE 与 A Level 表现的最强预测指标之一。两步线性方程是指对未知数恰好施加了两次逆运算的方程。本文将整个过程拆分为清晰步骤，涵盖天平法则、例题、常见错误、应用题，以及附完整答案的练习题。
1. What Is a Two-Step Equation? | 什么是两步方程？
In KS3 algebra, a linear equation is an equation in which the highest power of the unknown is 1. There are no squared terms, no cube terms, no square roots of the unknown and no unknown sitting in a denominator. The general form of a two-step equation is shown below, where a and b are numbers and x is the unknown.
在 KS3 代数中，线性方程是指未知数的最高次数为 1 的方程。它不含平方项、立方项，未知数不在根号里，也不在分母中。两步方程的一般形式如下，其中 a 和 b 是数字，x 是未知数。
ax + b = c
The phrase 'two-step' does not mean the solution is difficult. It simply tells you that two operations have been applied to the unknown, so you need to reverse two operations in the correct order. Once you can undo them in the right sequence, every question of this type becomes routine.
'两步' 这个词并不意味着解题很难。它只是告诉你未知数经历了两次运算，因此你需要按正确顺序反推两次运算。只要能按正确顺序反推，这类题目就会变得非常程式化。
The table below shows the four shapes a two-step equation usually takes. Reading them together helps you recognise the type quickly in an exam.
下表列出两步方程常见的四种形态。把它们放在一起看，有助于在考试中快速辨认题型。
| Form | Example | What happens to x |
| ax + b = c | 3x + 4 = 19 | multiplied, then a number added |
| ax − b = c | 5x − 7 = 23 | multiplied, then a number subtracted |
| b − ax = c | 12 − 2x = 4 | multiplied by a negative, then added to a number |
| x/3 + b = c | x/3 + 5 = 9 | divided, then a number added |
All four shapes are solved by the same two ideas: keep the equation balanced, and undo the operations in reverse order.
这四种形态都用同样的两个思想来解：保持方程平衡，并按相反顺序撤销运算。
2. Understanding the Balance Rule | 理解天平法则
All equation solving in KS3 depends on the balance rule. Think of each side of the equation as a pan on an old-fashioned balance scale. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side. This keeps the equation true while changing its form.
KS3 中所有解方程都依赖于天平法则。把方程的两边想象成老式天平的两个托盘。如果你对一边加、减、乘或除以一个数，必须对另一边做完全相同的操作。这样可以在保持方程成立的同时改变方程的形式。
ract 4 from both sides because 4 is being added on the left. The equation remains balanced.
例如，如果 3x + 4 = 19，你可以两边同时减去 4，因为左边加上了 4。方程保持平衡。
3x + 4 − 4 = 19 − 4 → 3x = 15
The new equation has exactly the same solution as the original one. Equations related in this way are called equivalent equations, and producing them is the whole foundation of algebra. An operation performed on one side only breaks the balance and will give a wrong answer.
新方程与原方程有完全相同的解。这样互相等价的方程称为同解方程，而制造同解方程正是整个代数学的基础。只对一边做运算会破坏平衡，必然得到错误答案。
A useful habit is to write the operation you are applying in the margin, for example '−4' on both sides. In an exam this also earns method marks even if a later arithmetic slip occurs.
一个有用的习惯是把你要做的运算写在旁边，例如在两边标注 '−4'。在考试中，即使后面出现算术失误，这样做也能拿到过程分。
3. The Correct Order of Inverse Operations | 逆运算的正确顺序
When you evaluate 3x + 4 for a given value of x, you multiply first and add second, following the order of operations. To solve, you reverse the process, which means undoing the last operation first: subtract 4, then divide by 3.
当你代入某个 x 值去计算 3x + 4 时，你会先乘后加，遵循运算顺序。解方程时要逆推这个过程，也就是先撤销最后一步运算：先减 4，再除以 3。
Undo + and − first, then undo × and ÷
Reversing in the wrong order is
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