📚 Solving Linear Equations | 解一元一次方程
Linear equations are the foundation of algebra at KS3. They appear in nearly every Cambridge Checkpoint test, often in question sets such as Exercise 72.2. Mastering how to balance and solve them gives you the confidence to tackle harder topics like inequalities, straight-line graphs and simultaneous equations. This article breaks down the key methods and typical exam-style questions step by step.
一元一次方程是 KS3 代数的核心基础。它们在剑桥 Checkpoint 考试中几乎随处可见，例如练习题 72.2 中的题型。掌握方程的平衡变形与求解方法，能帮助你更有信心地应对不等式、函数图像和联立方程等更难的课题。本文将分步拆解核心方法与典型考题。
1. What is a Linear Equation? | 什么是一元一次方程？
power of 1. For example, 3x + 2 = 14 is linear because x is not squared or cubed. The word 'linear' refers to the fact that its graph is a straight line.
一元一次方程是指未知数的指数只为 1 的方程。例如 3x + 2 = 14 就是线性方程，因为 x 没有平方或立方。“线性”一词来源于它的图像是一条直线。
- 3x + 2 = 14 — 3x 加 2 等于 14
- x − 7 = 10 — x 减 7 等于 10
- 2(x + 5) = 22 — 2 乘以括号 x 加 5 等于 22
2. The Balancing Method | 天平平衡法
Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side. This keeps the equation true. The goal is to isolate the variable on one side of the equals sign.
把方程想象成一台天平。无论你对等式的一边做什么，都必须对另一边做同样的操作，这样才能保持等式成立。我们的目标是把未知数单独留在等式的一边。
3x + 2 = 14
3x + 2 − 2 = 14 − 2 → 3x = 12 → x = 4
3. Solving One-Step Equations | 解一步方程
One-step equations need only one inverse operation. To undo addition, subtract; to undo subtraction, add; to undo multiplication, divide; to undo division, multiply.
一步方程只需要一次逆运算即可求解。要消去加法，就做减法；消去减法，就做加法；消去乘法，就做除法；消去除法，就做乘法。
x + 7 = 15 → x = 15 − 7 = 8
5x = 35 → x = 35 ÷ 5 = 7
4. Solving Two-Step Equations | 解两步方程
Two-step equations require two operations. Usually you first add or subtract the constant term, then multiply or divide by the coefficient of x. Write one operation per line to keep your working clear.
两步方程需要两次运算。通常先加减常数项，再对 x 的系数进行乘除。每行只写一步运算，保持解题过程清晰。
4x − 5 = 11
4x = 11 + 5 → 4x = 16 → x = 4
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive law. Multiply every term inside the bracket by the number outside. Then solve using the balancing method.
当方程含有括号时，先用分配律展开括号。把括号内的每一项都乘以外面的数，再用平衡法求解。
3(x − 2) = 12
3x − 6 = 12 → 3x = 18 → x = 6
A common error is to write 3(x − 2) as 3x − 2. Always multiply both x and −2 by 3.
一个常见错误是把 3(x − 2) 写成 3x − 2。务必把 x 和 −2 都乘以 3。
6. Variables on Both Sides | 未知数在等式两边
Some equations have x on both sides. Collect all variable terms on one side and all constant terms on the other side. Choose the side that keeps the coefficient of x positive if possible.
有些方程两边都有 x。需要把所有含未知数的项移到一边，把所有常数项移到另一边。尽量让 x 的系数保持为正。
5x + 3 = 2x + 12
5x − 2x = 12 − 3 → 3x = 9 → x = 3
7. Common Mistakes to Avoid | 常见错误与避坑指南
Many marks are lost from careless mistakes. Here are the top errors to watch out for when solving linear equations.
很多失分来自粗心错误。以下是最常见的几类错误，需要格外留意。
- Forgetting to apply an operation to both sides — 忘记对等式两边同时进行同一操作。
- Expanding brackets incorrectly, e.g. 3(x − 2) = 3x − 2 — 括号展开错误，例如 3(x − 2) 误写成 3x − 2。
- Sign errors when moving terms across the equals sign — 移项时符号错误。
- Dividing by the wrong number — 除以了错误的数。
- Not checking the final answer — 没有检验最终答案。
8. Word Problems into Equations | 应用题转化为方程
ibe a real-life situation. Convert the words into an equation using a variable. Key words: 'more than' means +, 'less than' means −, 'times' means ×, 'is' means =.
考试题经常描述一个实际情境。需要用变量把文字转化为方程。关键词：“比……多”表示加，“比……少”表示减，“……倍”表示乘，“是/等于”表示等号。
Example: Three more than twice a number is 19. Let the number be x.
例题：一个数的两倍再加 3 等于 19。设这个数为 x。
2x + 3 = 19
2x = 16 → x = 8
9. Quick Practice Set | 快速练习
Try these questions. Solutions are provided in the table below for self-check.
尝试以下练习题。答案附在下方表格中供自查。
| Question | 题目 | Answer | 答案 |
|---|---|
| 1. x + 9 = 17 （x 加 9 等于 17） | x = 8 （x 等于 8） |
| 2. 6x = 54 （6x 等于 54） | x = 9 （x 等于 9） |
| 3. 2x − 4 = 10 （2x 减 4 等于 10） | x = 7 （x 等于 7） |
| 4. 4(x + 1) = 24 （4 乘以括号 x 加 1 等于 24） | x = 5 （x 等于 5） |
| 5. 7x − 3 = 3x + 17 （7x 减 3 等于 3x 加 17） | x = 5 （x 等于 5） |
10. Exam Tips and Summary | 考试技巧与总结
Always show your working. Even if the final answer is wrong, method marks can be awarded. Substitute your answer back into the original equation to check. Keep equations balanced, and write one operation per line.
一定要展示解题步骤。即使最终答案错误，过程分也能拿到。把答案代回原方程进行检验。保持等式平衡，每行只写一步运算。
Remember: identify the inverse operations, apply them to both sides, expand brackets correctly, and check your answer. These skills will help you in KS3 tests and future Cambridge Checkpoint papers.
记住：确定逆运算，把它应用到等式两边，正确展开括号，并检验答案。这些技能将帮助你在 KS3 测试和未来的剑桥 Checkpoint 考试中取得好成绩。
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