📚 Solving Linear Equations: From Basics to Multi-Step Problems | 解一元一次方程：从基础到多步问题
Linear equations are one of the most important topics in the Cambridge KS3 mathematics curriculum. They appear in many forms, from simple one-step equations to multi-step problems that involve brackets, fractions, and variables on both sides. This article will guide you through the key skills step by step, with clear examples and exam-style questions.
一元一次方程是剑桥 KS3 数学课程中最重要的主题之一。它们以多种形式出现，从简单的一步方程到涉及括号、分数和两边变量的多步问题。本文将一步一步引导你掌握关键技能，配有清晰的例题和考试风格练习。
1. What is a Linear Equation? | 什么是一元一次方程？
A linear equation in one variable is an equation where the unknown appears only to the power of 1. For example, 3x + 5 = 20 is linear because x is not squared or cubed. The main goal is to find the value of the unknown that makes the equation true.
一元一次方程是指未知数只出现一次方（指数为 1）的方程。例如，3x + 5 = 20 是一元一次方程，因为 x 没有平方或立方。主要目标是找到使方程成立的未知数的值。
3x + 5 = 20
In Cambridge KS3, you are expected to solve equations like these using inverse operations on both sides, keeping the equation balanced at all times.
在剑桥 KS3 中，你需要通过在方程两边使用逆运算来解这类方程，并始终保持等式平衡。
2. Key Vocabulary and Symbols | 关键术语与符号
Understanding the language of algebra helps you follow instructions and show clear working. The table below lists common terms you will meet in linear equations.
理解代数语言有助于你理解题意并展示清晰的解题过程。下表列出你在解一元一次方程时会遇到的常见术语。
| Term | 术语 | Meaning | 含义 | Example | 例子 |
| Variable | 变量 | A letter that stands for an unknown number | 表示未知数的字母 | x in 3x + 5 |
| Coefficient | 系数 | The number in front of the variable | 变量前面的数字 | 3 in 3x |
| Constant | 常数 | A number on its own | 单独出现的数字 | 5 in 3x + 5 |
| LHS / RHS | 左边 / 右边 | Left-hand side and right-hand side of the equals sign | 等号的左边和右边 | 3x + 5 = 20 |
3. Solving One-Step Equations | 解一步方程
A one-step equation only needs one inverse operation to solve it. Remember to do the same thing to both sides.
一步方程只需要一个逆运算就可以解出。记住要对两边同时进行相同的运算。
x + 7 = 12
Subtract 7 from both sides: x = 5.
两边同时减去 7：x = 5。
x – 4 = 9
Add 4 to both sides: x = 13.
两边同时加 4：x = 13。
5x = 35
Divide both sides by 5: x = 7.
两边同时除以 5：x = 7。
x ÷ 3 = 6
Multiply both sides by 3: x = 18.
两边同时乘以 3：x = 18。
4. Solving Two-Step Equations | 解两步方程
Two-step equations require two inverse operations. The usual order is to undo addition or subtraction first, then undo multiplication or division.
两步方程需要两个逆运算。通常的顺序是先撤销加减法，再撤销乘除法。
2x + 3 = 11
Subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4.
两边同时减去 3：2x = 8。然后两边同时除以 2：x = 4。
7 – 2x = 1
First subtract 7 from both sides: -2x = -6. Then divide both sides by -2: x = 3.
首先两边同时减去 7：-2x = -6。然后两边同时除以 -2：x = 3。
5. Equations with Brackets | 含括号的方程
If an equation contains brackets, you can either expand them first or divide both sides by the factor in front of the brackets.
如果方程含有括号，你可以先展开括号，或者两边同时除以括号前的系数。
3(x + 4) = 27
Method 1: expand to get 3x + 12 = 27, then subtract 12: 3x = 15, so x = 5.
方法一：展开得到 3x + 12 = 27，然后减去 12：3x = 15，所以 x = 5。
Method 2: divide both sides by 3: x + 4 = 9, so x = 5.
方法二：两边同时除以 3：x + 4 = 9，所以 x = 5。
6. Equations with Variables on Both Sides | 变量在方程两边的情况
When the variable appears on both sides of the equals sign, collect the variable terms on one side and the constants on the other side.
当变量出现在等号两边时，把含有变量的项移到一边，把常数项移到另一边。
5x + 2 = 3x + 10
Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8, so x = 4.
两边同时减去 3x：2x + 2 = 10。然后再减去 2：2x = 8，所以 x = 4。
Always remember to change the sign when you move a term across the equals sign.
记得把项移过等号时要变号。
7. Equations with Fractions | 含分数的方程
Equations with fractions can be solved by multiplying every term by the lowest common denominator. This clears the fractions and creates a simpler equation.
含有分数的方程可以通过将每一项乘以最小公分母来解。这样可以消去分数，得到更简单的方程。
x/3 + 2 = 5
Multiply every term by 3: x + 6 = 15. Then subtract 6: x = 9.
每一项都乘以 3：x + 6 = 15。然后减去 6：x = 9。
(2x + 1)/3 = 5
Multiply both sides by 3: 2x + 1 = 15. Subtract 1: 2x = 14. Divide by 2: x = 7.
两边同时乘以 3：2x + 1 = 15。减去 1：2x = 14。除以 2：x = 7。
8. Word Problems Leading to Linear Equations | 由应用题建立线性方程
Many exam questions describe a situation in words. The key is to translate the words into an equation using a letter for the unknown.
许多考试题目用文字描述一个情境。关键是把文字翻译成方程，用一个字母表示未知数。
Example: The sum of twice a number and 7 is 25.
例子：一个数的两倍与 7 的和是 25。
Let the number be n. Then 2n + 7 = 25. Subtract 7: 2n = 18. Divide by 2: n = 9.
设这个数为 n。则 2n + 7 = 25。减去 7：2n = 18。除以 2：n = 9。
2n + 7 = 25
Always define your variable clearly at the start of your working.
在解题开始时始终清晰地定义你的变量。
9. Checking Your Solution | 检验你的解
After finding a solution, substitute it back into the original equation to check that both sides are equal.
求出解后，把它代回原方程，检查两边是否相等。
3x + 5 = 26
If x = 7: LHS = 3 × 7 + 5 = 26, RHS = 26. Since LHS = RHS, the solution is correct.
如果 x = 7：左边 = 3 × 7 + 5 = 26，右边 = 26。因为左边等于右边，所以解是正确的。
10. Common Mistakes to Avoid | 常见错误与避免方法
There are several common errors that students make when solving linear equations. Watch out for these in your own work.
学生在解一元一次方程时常犯以下几种错误。在你自己的作业中要注意这些。
Mistake 1: forgetting to change the sign when moving a term.
错误一：移项时忘记变号。
Mistake 2: multiplying only one term by the common denominator.
错误二：只把一项乘以公分母，而没有乘以所有项。
Mistake 3: expanding brackets incorrectly, such as writing 3(x + 4) = 3x + 4.
错误三：展开括号时出错，例如把 3(x + 4) 写成 3x + 4。
Mistake 4: not doing the same operation on both sides.
错误四：没有在两边进行相同的运算。
11. Practice Questions with Model Answers | 练习题与参考答案
Try these questions before looking at the answers. Cover the answers and write out full working for each one.
在看答案之前先尝试做这些题目。遮住答案，写出每道题的完整解题过程。
| Question | 题目 | Answer | 答案 |
| 1. x + 9 = 15 | x = 6 |
| 2. 4x = 28 | x = 7 |
| 3. 2x – 5 = 9 | x = 7 |
| 4. 3(x – 2) = 18 | x = 8 |
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