📚 Solving Linear Equations for KS3 Cambridge Maths | KS3 剑桥数学：解一元一次方程
Linear equations are the foundation of algebra at Key Stage 3. Whether you are working through the Cambridge Lower Secondary Maths course or preparing for a Checkpoint test, being able to solve an equation confidently is one of the most important skills you can practise. This article explains the key methods step by step, with examples, common errors and practice questions.
一元一次方程是 KS3 代数的基础。无论你正在学习剑桥初中数学课程，还是准备 Checkpoint 测试，能够自信地解方程都是你可以练习的最重要技能之一。本文将逐步解释关键方法，配以例题、常见错误和练习题。
1. What is a Linear Equation? | 什么是一元一次方程？
A linear equation is an equation where the unknown (usually written as x, y or n) is only raised to the power of 1. There are no x², x³, square roots or variables multiplied together. For example, 2x + 3 = 11 is a linear equation.
一元一次方程是指未知数（通常写作 x、y 或 n）的指数仅为 1 的方程。方程中不含 x²、x³、平方根，也不含变量相乘的形式。例如，2x + 3 = 11 就是一个一元一次方程。
The word ‘linear’ comes from ‘line’ because the graph of y = 2x + 3 is a straight line. Solving the equation tells us the value of the unknown that makes the statement true.
“线性” 一词来源于 “直线”，因为 y = 2x + 3 的图像是一条直线。解方程就是求出使等式成立的未知数的值。
2x + 3 = 11 is true when x = 4 because 2 × 4 + 3 = 11
2x + 3 = 11 在 x = 4 时成立，因为 2 × 4 + 3 = 11
2. Key Vocabulary and Notation | 关键术语与符号
Before solving equations, you need to be confident with the language used in questions. The table below shows the most common terms you will meet in Cambridge KS3 maths.
在解方程之前，你需要熟悉题目中常用的数学语言。下表列出了剑桥 KS3 数学中最常遇到的术语。
| English term | 中文术语 | Example |
| Variable | 变量 | x in 3x + 2 |
| Coefficient | 系数 | 3 in 3x + 2 |
| Constant | 常数 | 2 in 3x + 2 |
| Left-hand side (LHS) | 等号左边 | 2x + 3 in 2x + 3 = 11 |
| Right-hand side (RHS) | 等号右边 | 11 in 2x + 3 = 11 |
| Solution | 解 | x = 4 |
An equation is different from an expression. An expression such as 3x + 2 does not have an equals sign, so it cannot be solved directly. An equation such as 3x + 2 = 14 has an equals sign and can be solved to find the value of x.
方程与表达式不同。表达式如 3x + 2 没有等号，因此不能直接求解。方程如 3x + 2 = 14 有等号，可以求出 x 的值。
3. The Balance Method | 天平法原理
Think of an equation as a pair of balanced scales. The equals sign is the pivot. Whatever you do to one side, you must do exactly the same to the other side to keep the scales balanced.
把方程想象成一台平衡的天平。等号就是支点。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，才能保持天平平衡。
You can add, subtract, multiply or divide both sides by the same non-zero number. The aim is to isolate the unknown on one side of the equals sign.
你可以对两边同时加上、减去、乘以或除以同一个不为零的数。目标是把未知数单独留在等号的一边。
If x + 3 = 7, then x + 3 − 3 = 7 − 3, so x = 4
如果 x + 3 = 7，那么 x + 3 − 3 = 7 − 3，所以 x = 4
4. Solving One-Step Equations | 解一步方程
One-step equations need just one operation to find the unknown. Start with addition and subtraction, then move on to multiplication and division.
一步方程只需要一次运算就能求出未知数。先练习加法和减法，再练习乘法和除法。
For x + 5 = 13, subtract 5 from both sides to get x = 8. For x − 7 = 20, add 7 to both sides to get x = 27.
对于 x + 5 = 13，两边同时减去 5，得到 x = 8。对于 x − 7 = 20，两边同时加上 7，得到 x = 27。
x + 5 = 13 → x = 13 − 5 → x = 8
x − 7 = 20 → x = 20 + 7 → x = 27
For 3x = 27, divide both sides by 3 to get x = 9. For x ÷ 4 = 6, multiply both sides by 4 to get x = 24.
对于 3x = 27，两边同时除以 3，得到 x = 9。对于 x ÷ 4 = 6，两边同时乘以 4，得到 x = 24。
3x = 27 → x = 27 ÷ 3 → x = 9
x ÷ 4 = 6 → x = 6 × 4 → x = 24
5. Solving Two-Step Equations | 解两步方程
Two-step equations involve two operations. Always reverse the order of operations: undo addition or subtraction first, then undo multiplication or division.
两步方程包含两种运算。始终按运算的逆序求解：先消去加法或减法，再消去乘法或除法。
Solve 2x + 3 = 11. First subtract 3 from both sides to get 2x = 8. Then divide both sides by 2 to get x = 4.
解方程 2x + 3 = 11。首先两边同时减去 3，得到 2x = 8。然后两边同时除以 2，得到 x = 4。
2x + 3 = 11 → 2x = 11 − 3 → 2x = 8 → x = 8 ÷ 2 → x = 4
2x + 3 = 11 → 2x = 11 − 3 → 2x = 8 → x = 8 ÷ 2 → x = 4
Solve 5x − 4 = 21. Add 4 to both sides to get 5x = 25. Then divide both sides by 5 to get x = 5.
解方程 5x − 4 = 21。两边同时加上 4，得到 5x = 25。然后两边同时除以 5，得到 x = 5。
5x − 4 = 21 → 5x = 21 + 4 → 5x = 25 → x = 25 ÷ 5 → x = 5
5x − 4 = 21 → 5x = 21 + 4 → 5x = 25 → x = 25 ÷ 5 → x = 5
6. Equations with Brackets | 含括号的方程
If an equation contains brackets, expand them first using the distributive law. Then solve the resulting equation in the usual way.
如果方程中含有括号，先用分配律展开括号。然后按通常方法解得到的方程。
Solve 3(x + 2) = 18. Expand the left-hand side to get 3x + 6 = 18. Subtract 6 from both sides to get 3x = 12. Divide by 3 to get x = 4.
解方程 3(x + 2) = 18。展开左边得到 3x + 6 = 18。两边同时减去 6，得到 3x = 12。再除以 3，得到 x = 4。
3(x + 2) = 18 → 3x + 6 = 18 → 3x = 12 → x = 4
3(x + 2) = 18 → 3x + 6 = 18 → 3x = 12 → x = 4
Solve 2(3x − 1) = 16. Expand to get 6x − 2 = 16. Add 2 to both sides to get 6x = 18. Divide by 6 to get x = 3.
解方程 2(3x − 1) = 16。展开得到 6x − 2 = 16。两边同时加上 2，得到 6x = 18。再除以 6，得到 x = 3。
2(3x − 1) = 16 → 6x − 2 = 16 → 6x = 18 → x = 3
2(3x − 1) = 16 → 6x − 2 = 16 → 6x = 18 → x = 3
7. Equations with Unknowns on Both Sides | 未知数在等号两边的方程
When the unknown appears on both sides of the equals sign, collect the variable terms on one side and the constant terms on the other side. A good rule is to eliminate the smaller coefficient of x first.
当未知数出现在等号两边时，把含未知数的项移到一边，把常数项移到另一边。一个常用的方法是先消去系数较小的 x 项。
Solve 5x + 2 = 3x + 10. Subtract 3x from both sides to get 2x + 2 = 10. Subtract 2 from both sides to get 2x = 8. Divide by 2 to get x = 4.
解方程 5x + 2 = 3x + 10。两边同时减去 3x，得到 2x + 2 = 10。两边同时减去 2，得到 2x = 8。再除以 2，得到 x = 4。
5x + 2 = 3x + 10 → 2x + 2 = 10 → 2x = 8 → x = 4
5x + 2 = 3x + 10 → 2x + 2 = 10 → 2x = 8 → x = 4
Solve 7x − 5 = 2x + 20. Subtract 2x from both sides to get 5x − 5 = 20. Add 5 to both sides to get 5x = 25. Divide by 5 to get x = 5.
解方程 7x − 5 = 2x + 20。两边同时减去 2x，得到 5x − 5 = 20。两边同时加上 5，得到 5x = 25。再除以 5，得到 x = 5。
7x − 5 = 2x + 20 → 5x − 5 = 20 → 5x = 25 → x = 5
7x − 5 = 2x + 20 → 5x − 5 = 20 → 5x = 25 → x = 5
8. Equations Involving Fractions | 涉及分数的方程
When an equation includes a fraction, you can remove the fraction by multiplying both sides by the denominator. If there is more than one fraction, multiply by the lowest common denominator.
当方程中含有分数时，可以将两边同时乘以分母来消去分数。如果方程中有多个分数，则乘以它们的最小公分母。
Solve x ÷ 3 + 2 = 5. Subtract 2 from both sides to get x ÷ 3 = 3. Multiply both sides by 3 to get x = 9.
解方程 x ÷ 3 + 2 = 5。两边同时减去 2，得到 x ÷ 3 = 3。两边同时乘以 3，得到 x = 9。
x ÷ 3 + 2 = 5 → x ÷ 3 = 3 → x = 3 × 3 → x = 9
x ÷ 3 + 2 = 5 → x ÷ 3 = 3 → x = 3 × 3 → x = 9
Solve (2x) ÷ 5 − 1 = 7. Add 1 to both sides to get (2x) ÷ 5 = 8. Multiply both sides by 5 to get 2x = 40. Divide by 2 to get x = 20.
解方程 (2x) ÷ 5 − 1 = 7。两边同时加上 1，得到 (2x) ÷ 5 = 8。两边同时乘以 5，得到 2x = 40。再除以 2，得到 x = 20。
(2x) ÷ 5 − 1 = 7 → (2x) ÷ 5 = 8 → 2x = 40 → x = 20
(2x) ÷ 5 − 1 = 7 → (2x) ÷ 5 = 8 → 2x = 40 → x = 20
9. Checking Your Solution | 检验你的解
Always substitute your answer back into the original equation to check that it works. This is especially important in tests, because it can catch small mistakes quickly.
始终将你的答案代回原方程进行检验。这在考试中尤其重要，因为可以快速发现小的错误。
For 2x + 3 = 11, if x = 4, the left-hand side is 2 × 4 + 3 = 8 + 3 = 11. The right-hand side is 11. Both sides match, so the solution is correct.
对于 2x + 3 = 11，如果 x = 4，左边为 2 × 4 + 3 = 8 + 3 = 11。右边为 11。两边相等，因此解是正确的。
Check: 2 × 4 + 3 = 11 → 8 + 3 = 11 → 11 = 11 ✔
检验：2 × 4 + 3 = 11 → 8 + 3 = 11 → 11 = 11 ✔
10. Common Mistakes to Avoid | 常见错误与避免方法
Many students lose marks on algebra not because they do not understand the topic, but because they make small avoidable errors. Watch out for the following common mistakes.
许多学生在代数上丢分，并不是因为不懂知识点
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