📚 KS3 Cambridge Mathematics: Solving Linear Equations | 解一元一次方程
Linear equations are one of the most important building blocks in KS3 Cambridge Mathematics. They appear in number problems, geometry, statistics and real-life situations. Mastering how to solve an equation means learning how to keep both sides balanced while you find the unknown value. This article walks through the key methods step by step, from simple one-step equations to equations with brackets and unknowns on both sides.
一元一次方程是剑桥 KS3 数学中最重要的基础模块之一。它们出现在数字问题、几何、统计以及现实生活情境中。掌握解方程意味着学会在保持两边平衡的同时求出未知数的值。本文从简单的一步方程开始，逐步讲解含有括号和两边都含未知数的方程的关键方法。
1. What is a Linear Equation? | 什么是一元一次方程？
statement that contains an unknown value, often written as x, and an equals sign. The word 'linear' near equation because the highest power of x is 1. The goal is to find the value of x that makes the equation true.
一元一次方程是一个含有未知数（通常写作 x）和等号的代数陈述。“一次”表示未知数的最高次数为 1。例如 2x + 3 = 11 就是一个一元一次方程，因为 x 的最高次数是 1。我们的目标是求出使等式成立的 x 的值。
2. The Balancing Method | 平衡法
Think of an equation as a set of balance scales. The left side of the equals sign must always equal the right side. Whatever operation you do to one side, you must do exactly the same to the other side. If you add 4 to the left, add 4 to the right. If you divide the left by 3, divide the right by 3 too.
可以把方程想象成一台天平。等号左边必须始终等于等号右边。无论你对一边做什么运算，都必须对另一边做完全相同的运算。如果你在左边加上 4，右边也要加上 4。如果你把左边除以 3，右边也要除以 3。
x + 5 = 12 → x + 5 − 5 = 12 − 5 → x = 7
x + 5 = 12 → x + 5 − 5 = 12 − 5 → x = 7
3. Solving Equations with Addition and Subtraction | 用加法与减法解方程
s. If the equation is x − 8 = 3, add 8 to both sides. This leaves x by itself on one side.
当一个数加上或减去未知数时，使用逆运算将其移除。如果方程是 x + 6 = 10，两边同时减去 6。如果方程是 x − 8 = 3，两边同时加上 8。这样未知数 x 就单独留在了一边。
- Example 1: x + 9 = 20 → x = 20 − 9 → x = 11
- Example 2: x − 4 = 15 → x = 15 + 4 → x = 19
4. Solving Equations with Multiplication and Division | 用乘法与除法解方程
to isolate x. For example, 5x = 35 means x is multiplied by 5, so divide both sides by 5.
如果未知数乘以一个数，则两边同时除以这个数。如果未知数除以一个数，则两边同时乘以这个数。始终使用逆运算来分离 x。例如 5x = 35 表示 x 乘以 5，所以两边同时除以 5。
5x = 35 → 5x ÷ 5 = 35 ÷ 5 → x = 7
5x = 35 → 5x ÷ 5 = 35 ÷ 5 → x = 7
5. Two-Step Equations | 两步方程
irst and then divide by 3. Working in the correct order is essential because the unknown is involved in both operations.
许多方程需要两步才能解出。先消去加法或减法，再消去乘法或除法。对于 3x + 4 = 19，先减去 4，再除以 3。按照正确顺序求解非常重要，因为未知数同时参与这两种运算。
3x + 4 = 19 → 3x = 15 → x = 15 ÷ 3 → x = 5
3x + 4 = 19 → 3x = 15 → x = 15 ÷ 3 → x = 5
6. Expanding Brackets First | 先展开括号
nd divide by 2. Never try to divide both sides by 2 before expanding unless you also divide every term inside the bracket.
如果方程含有括号，在应用平衡法之前先展开括号。使用分配律：a(b + c) = ab + ac。对于 2(x + 3) = 14，先展开为 2x + 6 = 14，然后减去 6，再除以 2。除非括号内每一项都除以 2，否则不要在展开前先两边除以 2。
2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4
2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4
7. Equations with Unknowns on Both Sides | 两边都含未知数的方程
x + 14, subtract 2x from both sides to get 3x + 2 = 14, then solve as a two-step equation.
当未知数出现在方程两边时，把所有的 x 项移到一边，把所有的数字移到另一边。选择较小的 x 项来消去。对于 5x + 2 = 2x + 14，两边同时减去 2x 得到 3x + 2 = 14，然后按两步方程求解。
5x + 2 = 2x + 14 → 3x + 2 = 14 → 3x = 12 → x = 4
5x + 2 = 2x + 14 → 3x + 2 = 14 → 3x = 12 → x = 4
8. Equations Involving Fractions | 含有分数的方程
ides by 3 first, then subtract 2. This removes the fraction and makes the equation easier to handle.
当方程含有分数时，两边同时乘以分母以消去分数。如果 x/4 = 3，两边同时乘以 4。如果 (x + 2)/3 = 5，先两边同时乘以 3，再减去 2。这样可以消去分数，使方程更容易处理。
(x + 2)/3 = 5 → x + 2 = 15 → x = 13
(x + 2)/3 = 5 → x + 2 = 15 → x = 13
9. Checking Your Solution | 检验你的解
13, which is correct. This habit catches small arithmetic mistakes before they cost marks.
始终将你的答案代回原方程，确认它成立。如果你解 4x − 3 = 13 得到 x = 4，用 4 替换 x 进行检验：4 × 4 − 3 = 16 − 3 = 13，结果正确。这个习惯可以在小算术错误丢分之前将其发现。
10. Common Mistakes to Avoid | 常见错误与避免方法
- Forgetting to do the same operation to both sides: always keep the balance.
- Mixing up the order of inverse operations: undo addition before multiplication.
- Dividing before expanding brackets incorrectly: expand first unless every term is divided.
- Losing negative signs when moving terms: use inverse operations carefully.
- Not checking the solution: substitution prevents careless errors.
忘记对两边做相同的运算：必须始终保持平衡。混淆逆运算的顺序：先消去加法，再消去乘法。在展开括号前错误地做除法：除非每一项都除以该数，否则先展开。移项时丢失负号：仔细使用逆运算。不检验解：代入可以防止粗心错误。
11. Word Problems Leading to Linear Equations | 列方程解应用题
he unknown, choose a letter such as x, then translate the words into algebra. For example, "three more than twice a number is seventeen" becomes 2x + 3 = 17. Solve it and interpret the answer in context.
许多 KS3 题目要求你根据文字描述列出方程。确定未知数，选择一个字母如 x，然后把文字翻译成代数式。例如“一个数的两倍再加三等于十七”可以写成 2x + 3 = 17。解方程并结合题意解释答案。
2x + 3 = 17 → 2x = 14 → x = 7
2x + 3 = 17 → 2x = 14 → x = 7
12. Practice Questions and Summary | 练习题与小结
emember the five key steps: simplify both sides, collect x terms, collect number terms, divide to isolate x, then check by substitution.
尝试以下三道题来检验你的理解：解 x + 9 = 21，解 4(x − 2) = 24，解 7x − 5 = 3x + 19。记住五个关键步骤：化简两边、合并 x 项、合并数字项、除以系数分离 x，然后代入检验。
| Equation | Solution | Check |
|---|---|---|
| x + 9 = 21 | x = 12 | 12 + 9 = 21 |
| 4(x − 2) = 24 | x = 8 | 4(8 − 2) = 24 |
| 7x − 5 = 3x + 19 | x = 6 | 42 − 5 = 18 + 19 |
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