📚 Solving Linear Equations Step-by-Step | 线性方程求解步骤
Linear equations are one of the most important building blocks in KS3 mathematics. They show a balance between two expressions, and solving them means finding the unknown value that makes the statement true. This guide walks through the key methods you need for the Cambridge KS3 curriculum, from one-step equations to brackets and variables on both sides.
线性方程是 KS3 数学中最重要的基础内容之一。它们表示两个表达式之间的平衡，解方程就是求出让等式成立的未知数值。本指南将带你掌握剑桥 KS3 课程中需要的核心方法，从一步方程到含括号和两边含未知数的方程。
1. What Is a Linear Equation? | 什么是线性方程
ised to the power 1. For example, 2x + 3 = 11 is linear because x is not squared or cubed.
线性方程是指未知数（通常写作 x）的指数只为 1 的方程。例如 2x + 3 = 11 是线性方程，因为 x 没有平方或立方。
Linear equations can have one solution, no solution, or infinitely many solutions. At KS3 level, you will mostly meet equations with exactly one solution, and your task is to find that value of x.
线性方程可以有一个解、无解或无限多个解。在 KS3 阶段，你遇到的大多是恰好有一个解的方程，你的任务就是求出 x 的值。
2. The Golden Rule: Keep It Balanced | 黄金法则：保持平衡
Think of an equation as a set of scales. Whatever you do to one side, you must do to the other side. If you add 5 to the left, add 5 to the right. This keeps the scales balanced and the equation true.
把方程想象成一架天平。你对一边做的任何运算，都必须对另一边做同样的运算。如果左边加 5，右边也要加 5。这样才能保持天平平衡，方程仍然成立。
The table below shows the most common inverse operations used to keep equations balanced and isolate x.
下表展示了保持方程平衡并分离出 x 时最常用的逆运算。
| Operation | Inverse operation |
| Addition (+) | Subtraction (-) |
| Subtraction (-) | Addition (+) |
| Multiplication (×) | Division (÷) |
| Division (÷) | Multiplication (×) |
3. One-Step Equations | 一步方程
In a one-step equation, only one operation is needed to isolate x. Use the inverse operation: addition reverses subtraction, subtraction reverses addition, multiplication reverses division, and division reverses multiplication.
在一步方程中，只需要一步运算就能把 x 单独留在等号一边。使用逆运算：加法抵消减法，减法抵消加法，乘法抵消除法，除法抵消乘法。
x + 7 = 12 → x = 12 - 7 = 5
rom both sides to get x = 5. Always write the operation on both sides to show your method.
示例：x + 7 = 12。两边同时减去 7，得到 x = 5。一定要在两边同时写出运算，以展示你的解题步骤。
4. Two-Step Equations | 两步方程
Two-step equations involve two operations, such as multiplication and addition. Always undo the addition or subtraction first, then undo the multiplication or division.
两步方程包含两种运算，例如乘法和加法。一定要先消除加法或减法，再消除乘法或除法。
3x + 4 = 19 → 3x = 15 → x = 5
en divide both sides by 3 to find x = 5. Reversing the order is a common source of errors.
在上面的例子中，先两边减去 4，得到 3x = 15。然后两边除以 3，求出 x = 5。颠倒这个顺序是常见的错误来源。
5. Equations with Brackets | 含括号的方程
When brackets appear, expand them first using the distributive law. Multiply the term outside by every term inside the bracket before solving.
当方程中出现括号时，先用分配律展开。把括号外的项乘以括号内的每一项，然后再求解。
2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4
For 2(x + 3), the 2 multiplies both x and 3. After expanding, you have a two-step equation that is easier to solve using the method from Section 4.
对于 2(x + 3)，2 要同时乘以 x 和 3。展开之后，你就得到了一个两步方程，用第 4 节的方法更容易求解。
6. Variables on Both Sides | 两边含未知数的方程
If x appears on both sides, collect all x terms on one side and all number terms on the other side. You can do this by adding or subtracting the same x term from both sides.
如果 x 同时出现在等号两边，就把所有含 x 的项移到一边，把所有数字项移到另一边。你可以通过两边同时加上或减去同一个 x 项来实现。
5x - 3 = 2x + 9 → 5x - 2x = 9 + 3 → 3x = 12 → x = 4
In this example, subtract 2x from both sides to bring all x terms to the left, and add 3 to both sides to bring all numbers to the right. Then solve the final equation.
在这个例子中，两边同时减去 2x，把所有含 x 的项移到左边；两边同时加上 3，把所有数字移到右边。然后解最后的方程。
7. Checking Your Solution | 检验你的解
Always substitute your answer back into the original equation. If the left side equals the right side, your solution is correct. This check is quick and helps you catch small sign or arithmetic mistakes.
一定要把你的答案代回原方程。如果左边等于右边，你的解就是正确的。这个检验很快，能帮助你发现符号或计算方面的小错误。
then left side = 3 × 4 + 4 = 16, which is not 19, so you would know something went wrong.
例如，如果在 3x + 4 = 19 中得到 x = 4，那么左边 = 3 × 4 + 4 = 16，不等于 19，因此你就知道哪里出错了。
8. Common Mistakes to Avoid | 常见错误要避免
Below are some of the most frequent errors students make when solving linear equations. Watch out for them, especially in timed assessments.
以下是学生在解线性方程时最常犯的一些错误。尤其是在限时测验中，一定要格外留意。
- Forgetting to do the same operation on both sides.
- Dividing before subtracting in two-step equations.
- Mishandling negative signs.
- Expanding brackets incorrectly, for example writing 2(x + 3) as 2x + 3.
中文对照：忘记对两边做同样的运算；两步方程中先除后减；负号处理错误；括号展开错误，例如把 2(x + 3) 写成 2x + 3。
9. Word Problems | 应用题
Word problems require you to translate a sentence into an equation. Identify the unknown, choose a letter, and build the equation using key words such as 'total', 'difference', 'times' and 'less than'.
应用题要求你把文字叙述转化成方程。找出未知量，用一个字母表示，并利用 '总共'、'差'、'倍' 和 '比……少' 等关键词建立方程。
A number doubled and added to 5 gives 17: 2n + 5 = 17 → 2n = 12 → n = 6
Let the unknown number be n. The phrase 'doubled' means multiply by 2, and 'added to 5' gives + 5. After building the equation, solve it using the methods above.
设未知数为 n。'乘以 2' 表示 doubled，'再加 5' 表示 + 5。建立方程后，用前面的方法求解。
10. Quick Practice | 快速练习
Try these equations to check your understanding. Solve each one and then substitute your answer back into the original equation.
尝试解下列方程来检查你的理解。每解一个都要把答案代回原方程进行检验。
- x + 9 = 15
- 4x - 7 = 21
- 3(x - 2) = 18
- 7x + 4 = 3x + 24
参考答案依次为：x = 6, x = 7, x = 8, x = 5。如果你得到了不同的结果，请回看对应的小节并找出错误步骤。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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