📚 KS3 Cambridge Mathematics: Solving Linear Equations Step by Step | 剑桥初中数学：逐步解一元一次方程
Linear equations are one of the most important building blocks in KS3 Cambridge mathematics. This guide covers the key skills you need to solve one-step, two-step and multi-step equations with confidence.
一元一次方程是剑桥初中数学中最重要的基础内容之一。本指南涵盖了解一步、两步和多步方程所需的关键技能，帮助你建立信心。
1. What Is a Linear Equation? | 什么是一元一次方程？
+ 5 = 17. The unknown is usually written as a letter, often x, and its highest power is 1.
一元一次方程是指两个表达式相等的等式，例如 3x + 5 = 17。未知数通常用字母表示，常见为 x，且它的最高次数是 1。
2) = 28. The goal is always to find the value of the unknown that makes the equation true.
在剑桥初中阶段，你会遇到 x + 4 = 9、5x = 35、2x − 3 = 11 和 4(x + 2) = 28 等方程。目标始终是找出使等式成立的未知数的值。
Example: 3x + 5 = 17
例：3x + 5 = 17
2. The Balancing Method | 等式平衡法
Think of an equation as a pair of balanced scales. Whatever you do to one side, you must do exactly the same to the other side to keep the scales balanced.
把方程想象成一架平衡的天平。你对一边做的任何操作，都必须对另一边做完全相同的操作，才能保持天平平衡。
The two most common inverse operations are adding/subtracting the same number and multiplying/dividing by the same non-zero number.
两种最常见的逆运算是：同加或同减同一个数，以及同乘或同除以同一个非零数。
- If x + 3 = 10, subtract 3 from both sides: x = 7.
- 如果 x + 3 = 10，两边同时减 3：x = 7。
- If 2x = 14, divide both sides by 2: x = 7.
- 如果 2x = 14，两边同时除以 2：x = 7。
3. Solving One-Step Equations | 解一步方程
o isolate the unknown. For example, x + 6 = 13 is solved by subtracting 6 from both sides.
一步方程只需要一次逆运算就能把未知数单独求出来。例如 x + 6 = 13，两边同时减去 6 即可求解。
= 9, add 4 to both sides to get x = 13. For 5x = 45, divide both sides by 5 to get x = 9.
对于 x − 4 = 9，两边同时加 4，得到 x = 13。对于 5x = 45，两边同时除以 5，得到 x = 9。
x + 6 = 13 → x = 13 − 6 → x = 7
x + 6 = 13 → x = 13 − 6 → x = 7
4. Solving Two-Step Equations | 解两步方程
ultiplication and an addition. First subtract 3 from both sides, then divide both sides by 2.
两步方程包含两种运算。例如 2x + 3 = 15 含有乘法和加法。先用两边同时减 3，再两边同时除以 2。
The order matters: always undo addition or subtraction before undoing multiplication or division.
顺序很重要：一定要先处理加法或减法，再处理乘法或除法。
2x + 3 = 15 → 2x = 12 → x = 6
2x + 3 = 15 → 2x = 12 → x = 6
5. Unknown on Both Sides | 方程两边都有未知数
erms on one side and all the number terms on the other side. For example, 5x − 2 = 3x + 8.
当未知数同时出现在两边时，先把所有含未知数的项移到一边，把所有数字项移到另一边。例如 5x − 2 = 3x + 8。
t 3x from both sides to get 2x − 2 = 8. Then add 2 to both sides to get 2x = 10, so x = 5.
两边同时减 3x，得到 2x − 2 = 8。然后两边同时加 2，得到 2x = 10，因此 x = 5。
5x − 2 = 3x + 8 → 2x − 2 = 8 → 2x = 10 → x = 5
5x − 2 = 3x + 8 → 2x − 2 = 8 → 2x = 10 → x = 5
6. Expanding Brackets First | 先展开括号
hem before applying the balancing method. For example, 3(x + 4) = 27 becomes 3x + 12 = 27.
如果方程中含有括号，先展开括号，再使用等式平衡法。例如 3(x + 4) = 27 展开后得到 3x + 12 = 27。
Then subtract 12 from both sides to get 3x = 15, and divide by 3 to find x = 5.
然后两边同时减 12，得到 3x = 15，再除以 3，求出 x = 5。
3(x + 4) = 27 → 3x + 12 = 27 → 3x = 15 → x = 5
3(x + 4) = 27 → 3x + 12 = 27 → 3x = 15 → x = 5
7. Equations with Fractions | 含分数的方程
common multiple to clear the fractions. For example, x ÷ 4 = 3 can be written as x/4 = 3.
解含分数的方程时，把每一项都乘以分母或最小公倍数，以消去分数。例如 x ÷ 4 = 3 可以写作 x/4 = 3。
get x = 12. For (x + 2) ÷ 3 = 5, multiply both sides by 3 to get x + 2 = 15, then x = 13.
两边同时乘以 4，得到 x = 12。对于 (x + 2) ÷ 3 = 5，两边同时乘以 3，得到 x + 2 = 15，然后 x = 13。
x ÷ 4 = 3 → x = 12
x ÷ 4 = 3 → x = 12
8. Word Problems into Equations | 文字题转化为方程
Many exam questions describe a real-life situation. Read the problem carefully, let the unknown be x, and build an equation using the information given.
许多考试题目会描述一个实际情境。仔细读题，设未知数为 x，并根据已知信息建立方程。
Example: 'I think of a number, multiply it by 4 and add 7. The result is 31.' Let the number be x, so 4x + 7 = 31.
例：‘我想一个数，将它乘以 4，再加 7，结果是 31。’设这个数为 x，则 4x + 7 = 31。
Solving gives 4x = 24, so x = 6. Always answer the original question in words.
解得 4x = 24，因此 x = 6。最后一定要用文字回答原题。
9. Checking Your Solution | 检验你的解
After finding x, substitute it back into the original equation to check that both sides are equal. This is a quick and valuable exam habit.
求出 x 后，把它代回原方程，检查两边是否相等。这是一个快速且有价值的考试习惯。
+ 3 = 15, if x = 6, then left side = 2 × 6 + 3 = 12 + 3 = 15, which equals the right side.
对于 2x + 3 = 15，如果 x = 6，则左边 = 2 × 6 + 3 = 12 + 3 = 15，等于右边。
If the check fails, go back and look for an arithmetic or balancing error.
如果检验不成立，就返回去检查是否有计算或平衡操作的错误。
10. Common Mistakes | 常见错误
One common mistake is forgetting to apply an operation to both sides. Another is performing the wrong inverse operation, such as adding instead of subtracting.
一个常见错误是忘记对两边同时进行同一种操作。另一个错误是使用了错误的逆运算，例如该减却加了。
egative numbers. For 3x − 8 = 4, you must add 8 to both sides, giving 3x = 12, then x = 4.
还要注意负数。对于 3x − 8 = 4，必须两边同时加 8，得到 3x = 12，然后 x = 4。
- Do not move terms without changing signs correctly.
- 移项时不要忘记正确变号。
- Do not divide only part of one side.
- 不要只除以某一边的一部分。
11. Exam-Style Practice | 考试风格练习
Try these KS3 Cambridge style questions. Solve each equation and check your answer.
试试以下剑桥初中风格的题目。解出每个方程并检验答案。
- 4x + 9 = 33
- 7x − 5 = 2x + 20
- 2(x − 3) = 18
- x ÷ 5 + 2 = 7
re x = 6, x = 5, x = 12 and x = 25. Use the balancing method to show your working clearly.
答案分别是 x = 6、x = 5、x = 12 和 x = 25。使用平衡法清晰展示你的解题步骤。
12. Summary | 小结
To solve linear equations successfully, identify the operations involved, undo them in reverse order using inverse operations, and always keep both sides balanced.
要成功解一元一次方程，先识别涉及的运算，再用逆运算按相反顺序逐步化简，并始终保持两边平衡。
| Equation type | 方程类型 | Strategy | 策略 |
| One-step | 一步方程 | Use one inverse operation | 使用一次逆运算 |
| Two-step | 两步方程 | Undo addition/subtraction first | 先处理加减法 |
| Unknown on both sides | 两边有未知数 | Collect like terms | 合并同类项 |
| With brackets | 含括号 | Expand first | 先展开括号 |
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