📚 Mastering Linear Equations from p217_2.pdf | 掌握一元一次方程：p217_2.pdf 精讲
ge KS3 students revise linear equations. You will learn how to read the question, form an equation, solve it step by step, and check your answer using substitution.
本文以 p217_2.pdf 中出现的题型为切入点，帮助 Cambridge KS3 学生复习一元一次方程。你将学会如何理解题意、列出方程、逐步求解，并通过代入检验答案。
1. What the Question Asks | 题目要求
) = 2x + 8. It may also ask you to form an equation from a word problem and then solve it.
在许多 Cambridge KS3 试卷中，像 p217_2.pdf 这样的题目通常会要求你解一个一元一次方程，例如 4(x − 3) = 2x + 8。题目也可能要求你先根据文字题列出方程，再求解。
You must show each working step clearly. Marks are often awarded for your method, not only for the final answer.
你必须清楚写出每一步计算过程。评分标准通常会给方法分，而不只看最后的答案。
2. Key Ideas and Vocabulary | 关键概念与词汇
Before solving the question, make sure you know the main words and symbols used in algebra. A linear equation is an equation where the variable has no power higher than 1.
在解题之前，要确保你掌握代数中常用的词汇和符号。一元一次方程是指未知数的最高次数为 1 的方程。
The table below lists the key terms you need to recognise.
下表列出了你需要识别的重要术语。
| English Term | 中文术语 | Meaning |
|---|---|---|
| Variable | 变量 | A letter such as x that stands for an unknown number |
| Equation | 方程 | A mathematical statement with an equals sign |
| Solution | 解 | The value of the variable that makes the equation true |
| Expand | 展开 | To remove brackets by multiplying each term inside |
| Substitute | 代入 | To replace a variable with a number |
3. Writing Expressions from Words | 从文字列代数式
Many algebra questions start with a phrase such as 'three more than twice a number' or 'the perimeter of a rectangle'. You must translate these words into an algebraic expression.
许多代数题会先给出诸如“一个数的两倍再加三”或“长方形的周长”这样的文字描述。你必须把这些文字转换成代数式。
For example, if n is the unknown number, then 'twice a number' is written as 2n, and 'three more than twice a number' is written as 2n + 3.
例如，如果 n 表示未知数，那么“一个数的两倍”写作 2n，“一个数的两倍再加三”写作 2n + 3。
Always define your variable clearly before you form an equation. Writing 'Let x be...' helps you and the examiner follow your logic.
在列方程之前，一定要清楚地定义变量。写出“设 x 为……”可以帮助你和阅卷老师理解你的思路。
4. One-Step and Two-Step Equations | 一步与两步方程
ed in one step. For example, to solve x + 5 = 12, subtract 5 from both sides to get x = 7.
最简单的一元一次方程只需一步就能解出。例如，解 x + 5 = 12 时，两边同时减去 5，得到 x = 7。
operations. To solve 3x − 4 = 11, first add 4 to both sides, then divide both sides by 3.
两步方程需要进行两次运算。解 3x − 4 = 11 时，先在两边同时加上 4，再在两边同时除以 3。
3x − 4 = 11
3x = 15
x = 5
5. Brackets and Expansion | 去括号与展开
Equations often contain brackets. You must expand them before collecting like terms. The rule is to multiply each term inside the bracket by the number or letter outside.
方程中经常含有括号。在合并同类项之前，必须先展开括号。规则是用括号外面的数或字母去乘括号内的每一项。
For example, 4(x − 3) becomes 4x − 12 because 4 × x = 4x and 4 × (−3) = −12.
例如，4(x − 3) 展开后为 4x − 12，因为 4 × x = 4x，而 4 × (−3) = −12。
Be careful with negative signs. Expanding −2(x + 5) gives −2x − 10, not −2x + 10.
要特别注意负号。展开 −2(x + 5) 得到 −2x − 10，而不是 −2x + 10。
6. Variables on Both Sides | 两边含变量的方程
When an equation has variables on both sides, collect all variable terms on one side and all number terms on the other side.
当方程两边都含有变量时，要把所有含变量的项移到一边，把所有常数项移到另一边。
example, to solve 4x − 12 = 2x + 8, first subtract 2x from both sides to get 2x − 12 = 8.
例如，解 4x − 12 = 2x + 8 时，先在两边同时减去 2x，得到 2x − 12 = 8。
en add 12 to both sides to get 2x = 20, and finally divide both sides by 2 to find x = 10.
然后在两边同时加上 12，得到 2x = 20，最后两边同时除以 2，求出 x = 10。
7. Worked Example from p217_2.pdf | p217_2.pdf 典型例题
work through a complete example of the type found in p217_2.pdf: solve 4(x − 3) = 2x + 8.
现在我们来完整解一道 p217_2.pdf 中的典型例题：解方程 4(x − 3) = 2x + 8。
Step 1: Expand the bracket on the left-hand side.
第一步：展开左边的括号。
4x − 12 = 2x + 8
Step 2: Subtract 2x from both sides so that the variable terms sit on the left.
第二步：在两边同时减去 2x，使含变量的项都移到左边。
2x − 12 = 8
Step 3: Add 12 to both sides to isolate the term 2x.
第三步：在两边同时加上 12，使 2x 这一项单独在一边。
2x = 20
Step 4: Divide both sides by 2.
第四步：两边同时除以 2。
x = 10
You have now found the solution. Do not stop here; always check your answer.
你已经求出了方程的解。不要就此停下；一定要检验答案。
8. Checking Your Solution | 检验答案
titute x = 10 back into the original equation to make sure both sides have the same value.
把 x = 10 代回原方程，确认两边相等。
Left-hand side: 4(x − 3) = 4(10 − 3) = 4 × 7 = 28.
左边：4(x − 3) = 4(10 − 3) = 4 × 7 = 28。
Right-hand side: 2x + 8 = 2 × 10 + 8 = 20 + 8 = 28.
右边：2x + 8 = 2 × 10 + 8 = 20 + 8 = 28。
Since both sides equal 28, the solution x = 10 is correct.
因为两边都等于 28，所以 x = 10 这个解是正确的。
9. Common Errors to Avoid | 常见错误
One common mistake is forgetting to multiply every term inside a bracket. For example, writing 4(x − 3) as 4x − 3 instead of 4x − 12.
一个常见错误是忘记乘以括号内的每一项。例如把 4(x − 3) 写成 4x − 3，而不是 4x − 12。
Another mistake is changing the sign incorrectly when moving terms across the equals sign. Moving 2x to the left gives −2x, but moving −12 to the right gives +12.
另一个错误是在移项时符号处理不当。把 2x 移到左边会变成 −2x，而把 −12 移到右边会变成 +12。
To avoid these errors, write one operation per line and check each side of the equation stays balanced.
为了避免这些错误，每行只做一个运算，并检查方程两边始终保持平衡。
10. Practice and Exam Tips | 练习与考试技巧
Try these two quick questions on your own before looking at the hints.
在看提示之前，请先自己完成下面两道快速练习。
- Solve 3(2x − 4) = 18
- Solve 5x + 7 = 2x − 8
For the first question, expand the bracket, add 12 to both sides, then divide by 6. For the second question, subtract 2x from both sides, subtract 7, then divide by 3.
第一题先展开括号，两边加 12，再除以 6。第二题两边先减去 2x，再减去 7，最后除以 3。
In the exam, always show your method, write clearly, and leave time to substitute your answer back into the original equation.
考试时一定要展示解题方法，书写工整，并留出时间把答案代回原方程进行检验。
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