📚 Solving Linear Equations: KS3 Cambridge Maths | 解一元一次方程：KS3剑桥数学
common question styles and exam tips, closely matched to tasks such as those on p92_2.pdf.
解一元一次方程是 KS3 剑桥数学课程中最重要的技能之一。它以数的运算为基础，为学生进一步学习代数、图像和问题解决打下基础。本文回顾核心方法、常见题型和考试技巧，紧扣 p92_2.pdf 等典型练习。
1. What Is a Linear Equation? | 什么是线性方程？
example, 3x + 2 = 11 is a linear equation because the variable x is not squared or cubed.
线性方程是一个代数等式，其中两个表达式相等，未知数的次数仅为 1。例如 3x + 2 = 11 是线性方程，因为变量 x 没有被平方或立方。
In KS3, the main goal is to find the value of the unknown that makes the equation true. This value is called the solution or root.
在 KS3 阶段，主要目标是求出使等式成立的未知数的值。这个值称为方程的解或根。
2. The Balance Method | 天平法（等式平衡法）
The balance method treats an equation like a set of scales. Whatever you do to one side, you must do to the other side to keep the equation balanced.
天平法将方程视为一架天平。对等式一边施加任何运算，必须对另一边做同样运算，才能保持等式平衡。
For example, if you add 3 to the left side, you must also add 3 to the right side. If you divide the left side by 5, you must divide the right side by 5.
例如，如果在左边加 3，右边也必须加 3；如果将左边除以 5，右边也必须除以 5。
x + 3 = 10 → x + 3 - 3 = 10 - 3 → x = 7
3. Solving One-Step Equations | 解一步方程
A one-step equation needs only one inverse operation to find the solution. The four basic types are addition, subtraction, multiplication and division.
一步方程只需一步逆运算即可求出解。四种基本类型是加法、减法、乘法和除法。
| Type | Example | Solution |
|---|---|---|
| Addition | x + 4 = 9 | x = 5 |
| Subtraction | x - 6 = 3 | x = 9 |
| Multiplication | 5x = 20 | x = 4 |
| Division | x ÷ 3 = 7 | x = 21 |
Remember to use the inverse operation: addition undoes subtraction, multiplication undoes division, and vice versa.
记住使用逆运算：加法抵消减法，乘法抵消除法，反之亦然。
4. Solving Two-Step Equations | 解两步方程
A two-step equation involves two operations. The standard method is to undo the addition or subtraction first, then undo the multiplication or division.
两步方程包含两种运算。标准方法是先消去加或减，再消去乘或除。
2x + 5 = 17 → 2x = 12 → x = 6
versed if the variable is divided before a term is added: x/3 - 4 = 8 → x/3 = 12 → x = 36.
有时顺序相反，例如变量先除以 3 再减去 4：x/3 - 4 = 8 → x/3 = 12 → x = 36。
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive law, then simplify and solve as usual.
当方程含有括号时，先用分配律展开括号，然后化简并照常求解。
3(x + 2) = 21 → 3x + 6 = 21 → 3x = 15 → x = 5
er outside. A common sign error is to forget the negative sign, as in -2(x - 3) = -2x + 6.
必须用括号外的数乘以括号内的每一项。常见符号错误是忘记负号，例如 -2(x - 3) = -2x + 6。
6. Equations with Variables on Both Sides | 含两边变量的方程
If the unknown appears on both sides of the equation, first collect the variable terms on one side and the number terms on the other side.
如果未知数出现在等式两边，先将含变量的项集中到一边，将常数项集中到另一边。
5x + 3 = 2x + 12 → 3x = 9 → x = 3
Aim to keep the variable coefficient positive to reduce sign errors. In the above case, subtracting 2x from both sides is easier than subtracting 5x.
尽量保持变量系数为正，以减少符号错误。在上例中，两边减去 2x 比减去 5x 更容易。
7. Equations with Fractions | 含分数的方程
To solve equations involving fractions, multiply every term on both sides by the least common denominator (LCD) to clear the fractions.
解含分数的方程时，先将等式两边每一项都乘最小公分母（LCD），以消去分母。
x/4 + 1 = 3 → x/4 = 2 → x = 8
A more complex example: (2x)/3 = 4 → multiply both sides by 3: 2x = 12 → x = 6.
更复杂的例子：(2x)/3 = 4 → 两边乘以 3：2x = 12 → x = 6。
8. Translating Word Problems into Equations | 将文字题转化为方程
Many KS3 problems ask you to form and solve an equation from a real-life situation. Use a letter for the unknown and build the equation step by step.
许多 KS3 题目要求根据实际情境建立并解方程。用字母表示未知数，并逐步建立方程。
ly it by 4, then subtract 7. The answer is 25. Let n be the number: 4n - 7 = 25, so n = 8.
例如：我想一个数，将它乘以 4，再减去 7，结果是 25。设这个数为 n：4n - 7 = 25，因此 n = 8。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
The most common errors include doing an operation to only one side, forgetting to change signs when expanding brackets, and mixing up the order of inverse operations.
最常见的错误包括只对等式一边进行运算、展开括号时忘记变号、混淆逆运算的顺序。
Always use a written check by substituting your solution back into the original equation. This takes a few seconds and can catch many mistakes.
始终通过代入原方程进行书面检验。这只需几秒钟，却能发现许多错误。
10. Practice Questions and Checkpoint Tips | 练习题与 Checkpoint 应考提示
to consolidate your skills: 7x - 2 = 26, 4(x - 3) = 32, 3x + 5 = 2x + 17, and x/5 + 2 = 7.
尝试以下题目以巩固技能：7x - 2 = 26，4(x - 3) = 32，3x + 5 = 2x + 17，x/5 + 2 = 7。
In the Cambridge Checkpoint test, show every step clearly. Even if your final answer is wrong, method marks can be awarded for correct balance and inverse operations.
在剑桥 Checkpoint 测试中，要清楚地写出每一步。即使最终答案错误，正确的等式平衡和逆运算步骤也能获得方法分。
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