📚 Solving Linear Equations | 解线性方程
Linear equations are the foundation of algebra. At KS3, you learn to solve equations where the unknown (often x) appears only to the power of 1. Understanding how to isolate the variable step by step will help you tackle problems in geometry, science, and everyday reasoning. In this article, we will explore methods such as balancing, expanding brackets, and dealing with fractions, all while practising clear logical steps.
线性方程是代数的基础。在 KS3 阶段，你将学习求解未知数（通常是 x）的指数为 1 的方程。理解如何一步步分离变量将有助于你解决几何、科学以及日常推理中的问题。在本文中，我们将探索诸如平衡法、展开括号和处理分数等技巧，同时培养清晰的逻辑步骤。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is an equation where the highest power of the variable is 1. For example, 2x + 3 = 11 is a linear equation because x is not squared or cubed. The solution is the value of x that makes the equation true. Graphically, it represents a straight line.
线性方程是指变量的最高次幂为 1 的方程。例如，2x + 3 = 11 就是一个线性方程，因为 x 没有被平方或立方。方程的解就是使等式成立的 x 值。从图形上看，它表示一条直线。
2. The Balancing Method | 平衡法
Think of an equation as a set of scales. Whatever you do to one side, you must do to the other to keep the scales balanced. If you add 5 to the left, add 5 to the right. This principle allows you to systematically solve for x.
把方程想象成一台天平。你对一边所做的任何操作，都必须对另一边做同样的操作，以保持天平平衡。如果你在左边加 5，那么在右边也加 5。这条原则让你能够有条不紊地解出 x。
3. Solving by Inverse Operations | 利用逆运算求解
To isolate x, you undo the operations that are applied to it. For example, if x is multiplied by 4, you divide by 4. If 7 is subtracted, you add 7. Always apply inverse operations in the reverse order of BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction).
为了分离 x，你需要撤销对它施加的运算。例如，如果 x 乘以了 4，你就除以 4；如果减去了 7，你就加上 7。永远按照 BIDMAS（括号、指数、除法/乘法、加法/减法）的逆序来应用逆运算。
4. Example: Two-Step Equation | 示例：两步方程
Solve 3x – 7 = 5. Step 1: add 7 to both sides → 3x = 12. Step 2: divide both sides by 3 → x = 4. Check by substituting back: 3(4) – 7 = 12 – 7 = 5 ✓.
求解 3x – 7 = 5。第一步：两边加 7 → 3x = 12。第二步：两边除以 3 → x = 4。通过代回检验：3(4) – 7 = 12 – 7 = 5 ✓。
5. Equations with the Unknown on Both Sides | 未知数在两边的情况
When x appears on both sides, collect all x‑terms on one side and numbers on the other. For 5x + 2 = 3x + 10, subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8, so x = 4.
当 x 出现在两边时，将所有含 x 的项移到一边，数字移到另一边。对于 5x + 2 = 3x + 10，两边同时减去 3x：2x + 2 = 10，然后再减去 2：2x = 8，因此 x = 4。
5x + 2 = 3x + 10 → 2x = 8 → x = 4
6. Expanding Brackets First | 先展开括号
If an equation contains brackets, expand them before simplifying. Solve 2(3x – 1) = 4x + 6. Expand: 6x – 2 = 4x + 6. Subtract 4x: 2x – 2 = 6. Add 2: 2x = 8, so x = 4.
如果方程含有括号，先将其展开再化简。求解 2(3x – 1) = 4x + 6。展开得：6x – 2 = 4x + 6。减去 4x：2x – 2 = 6。加 2：2x = 8，故 x = 4。
7. Dealing with Fractions | 处理含有分数的方程
Multiply every term by the lowest common denominator (LCD) to eliminate fractions. For (x/3) + 1 = (x – 2)/2, multiply by 6: 2x + 6 = 3(x – 2). Then expand: 2x + 6 = 3x – 6. Subtract 2x and add 6: 12 = x, so x = 12.
将每一项乘以最小公分母（LCD）来消去分母。对于 (x/3) + 1 = (x – 2)/2，乘以 6 得：2x + 6 = 3(x – 2)。然后展开：2x + 6 = 3x – 6。减去 2x 并加 6：12 = x，因此 x = 12。
(x/3) + 1 = (x – 2)/2 → ×6 → 2x + 6 = 3x – 6 → x = 12
8. Checking Your Solution | 检验你的解
Always substitute your answer back into the original equation. If both sides are equal, your solution is correct. This habit catches arithmetic mistakes and builds confidence.
始终将你的答案代回原方程。如果两边相等，你的解就是正确的。这个习惯能发现计算错误并树立信心。
9. Word Problems Leading to Linear Equations | 可转化为线性方程的文字题
Translate the words into algebra: “I think of a number, double it and add 7. The result is 21.” Let the number be n → 2n + 7 = 21 → 2n = 14 → n = 7. The number is 7.
将文字转化成代数语言：“我想了一个数，把它加倍再加 7，结果是 21。” 设这个数为 n → 2n + 7 = 21 → 2n = 14 → n = 7。这个数就是 7。
- Identify the unknown and choose a letter.
- Form an equation based on the operations described.
- Solve the equation.
- Answer the question in words.
- 确定未知数并选定一个字母。
- 根据描述的运算建立方程。
- 求解方程。
- 用文字作答。
10. Common Mistakes to Avoid | 要避免的常见错误
Some learners forget to perform the same operation on both sides. Others incorrectly handle negative signs or fractions. Also, when expanding brackets like –2(x – 3), remember to multiply both terms: –2x + 6, not –2x – 6. Practise deliberately to minimise these errors.
有些学习者忘记对两边同时进行相同的操作。另一些人则不能正确处理负号或分数。还有，在展开像 –2(x – 3) 这样的括号时，记住要乘两项：得到 –2x + 6，而不是 –2x – 6。有意识地练习以尽量减少这些错误。
| Common Error / 常见错误 | Correction / 纠正 |
|---|---|
| Adding on one side only | Always add to both sides |
| Forgetting to multiply both terms when expanding | Multiply every term inside the bracket |
| Dividing by a negative incorrectly | Keep the sign: –2x = 6 → x = –3 |
| 只在一边加上 | 永远同时在两边加上 |
| 展开时忘记乘以括号内的每一项 | 括号内的每一项都乘 |
| 负数除法符号错误 | 保留符号：–2x = 6 → x = –3 |
11. Equations with Decimals | 含有小数的方程
You can clear decimals by multiplying by 10, 100, etc., just like with fractions. Solve 0.2x + 0.3 = 0.7. Multiply by 10: 2x + 3 = 7 → 2x = 4 → x = 2. Checking: 0.2(2) + 0.3 = 0.4 + 0.3 = 0.7 ✓.
你可以像处理分数那样，通过乘以 10、100 等来消去小数。求解 0.2x + 0.3 = 0.7。乘以 10 得：2x + 3 = 7 → 2x = 4 → x = 2。检验：0.2(2) + 0.3 = 0.4 + 0.3 = 0.7 ✓。
12. Summary and Practice Tips | 小结与练习建议
Mastering linear equations requires consistent practice. Start with simple one‑step equations, then progress to two‑step, brackets, unknowns on both sides, and finally fractions and decimals. Write down each step neatly, and always check your answer. Use online quizzes or textbook exercises to build fluency.
掌握线性方程需要持之以恒的练习。从简单的一步方程开始，然后进阶到两步方程、含括号的、未知数在两边的情况，最后是分数和小数。把每一步工整地写下来，并且一定要检验答案。利用在线测验或课本习题来提高熟练度。
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