📚 Solving Linear Equations | 解一元一次方程
mistakes and exam-style examples, including the skills practised on pages such as p203_2.
线性方程是 KS3 代数的核心基础。在剑桥初中数学中，你需要会用逆运算建立、化简并求解一元方程，还要检验答案。本文梳理关键方法、常见错误和考试风格例题，涵盖类似 p203_2 练习页中的技能。
1. What Is a Linear Equation? | 什么是线性方程？
tions. There are no x² terms, no x³ terms and no products like xy. Linear equations are called linear because their graphs are straight lines when plotted on a coordinate plane, even though at KS3 we mostly solve them algebraically.
一元线性方程是指未知数（通常用 x 表示）的最高次数为 1 的方程。它可以写成 ax + b = c、ax + b = cx + d，也可以含有括号和分数。方程中不能出现 x²、x³ 或 xy 这样的项。线性方程之所以叫线性，是因为在坐标平面上画出来是一条直线，不过在 KS3 阶段我们主要用代数方法求解。
ax + b = c
2. The Balance Method | 天平法
You can think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep the equation true. Adding, subtracting, multiplying or dividing both sides by the same non-zero number is allowed. This idea is the foundation of every correct step in solving a linear equation.
你可以把方程想象成一架天平。对一边做了什么，另一边也必须做同样的事，才能保持方程成立。两边同时加、减、乘或除以同一个非零数是可以的。这个思想是解线性方程每一步正确操作的基础。
if x + 4 = 9, you can subtract 4 from both sides because the two sides must remain equal.
例如，如果 x + 4 = 9，你可以两边同时减去 4，因为两边必须始终保持相等。
x + 4 − 4 = 9 − 4
x = 5
3. Solving Basic Equations | 解基本方程
sides by a. These four operations are the reverse of the operations that are applied to x.
解 x + a = b 时两边同时减去 a；解 x − a = b 时两边同时加上 a；解 ax = b 时两边同时除以 a；解 x/a = b 时两边同时乘以 a。这四种运算就是把作用在 x 上的运算倒过来。
| Equation type | Operation | Example |
|---|---|---|
| x + a = b | Subtract a from both sides | x + 7 = 15 → x = 8 |
| x − a = b | Add a to both sides | x − 3 = 10 → x = 13 |
| ax = b | Divide both sides by a | 5x = 35 → x = 7 |
| x/a = b | Multiply both sides by a | x/3 = 4 → x = 12 |
Always write every step clearly, especially when negative signs are involved.
每一步都要写清楚，尤其是涉及负号的时候。
4. Collecting Like Terms First | 先合并同类项
Before solving, collect like terms on each side. For example, 3x + 2x becomes 5x, and 8 − 3 becomes 5. Simplifying first makes later steps much easier and reduces sign errors. Always check that the terms you combine have exactly the
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