Mastering Linear Equations for KS3 Cambridge Mathematics | KS3 剑桥数学：掌握线性方程

📚 Mastering Linear Equations for KS3 Cambridge Mathematics | KS3 剑桥数学：掌握线性方程

Linear equations are one of the most important building blocks in KS3 Cambridge Mathematics. They appear in every topic from geometry to statistics, and mastering them early will make Year 9 and IGCSE work much easier. This article walks you through the key ideas, methods and common mistakes step by step, with practice questions at the end.

线性方程是 KS3 剑桥数学中最重要的基础模块之一。从几何到统计，它们几乎出现在每个主题中；尽早掌握线性方程，会让 Year 9 和 IGCSE 的学习轻松很多。本文逐步讲解核心概念、解题方法和常见错误，并在最后配有练习题。

1. What is a Linear Equation? | 什么是线性方程？

A linear equation is a statement that two expressions are equal, and the unknown appears only to the power of 1. The word ‘linear’ means that if you draw its graph, you get a straight line. Examples include x + 5 = 12, 3x – 7 = 8 and 2(x + 4) = 18.

线性方程是表示两个表达式相等的一种陈述，未知数的次数仅为 1 次。”线性”这个词意味着如果你画出它的图像，会得到一条直线。例如 x + 5 = 12、3x – 7 = 8 和 2(x + 4) = 18。

linear equation: ax + b = c, where a ≠ 0

线性方程：ax + b = c，其中 a ≠ 0

In KS3 you will mostly solve equations where the unknown is called x, y or n. The goal is always the same: find the value of the unknown that makes the equation true.

在 KS3 阶段，你主要会解未知数用 x、y 或 n 表示的方程。目标始终相同：求出使方程成立的未知数的值。

2. The Balancing Method | 等式的平衡法

Think of an equation as a balance scale. Whatever you do to one side, you must do exactly the same to the other side, otherwise the scale tips. This is the golden rule of solving equations.

把方程想象成一架天平。无论你对一边做什么运算，另一边也必须做完全相同的运算，否则天平就会倾斜。这是解方程的黄金法则。

Add the same number to both sides: both sides add the same number
Subtract the same number from both sides: both sides subtract the same number
Multiply or divide both sides by the same non-zero number: both sides multiply or divide by the same non-zero number

两边同时加上同一个数
两边同时减去同一个数
两边同时乘以或除以同一个非零的数

For example, if x + 3 = 10, we can subtract 3 from both sides to keep the equation balanced: x + 3 – 3 = 10 – 3, so x = 7.

例如，如果 x + 3 = 10，我们可以从两边同时减去 3 来保持等式平衡：x + 3 – 3 = 10 – 3，因此 x = 7。

3. Solving One-Step Equations | 解一步方程

One-step equations can be solved by performing one inverse operation. If x has been increased by a number, subtract that number from both sides. If x has been multiplied by a number, divide both sides by that number.

一步方程只需使用一次逆运算即可解出。如果 x 被加上了一个数，就在两边同时减去这个数。如果 x 被乘以了一个数，就在两边同时除以这个数。

Example 1: Solve x + 8 = 15.

示例 1：解方程 x + 8 = 15。

x + 8 – 8 = 15 – 8 => x = 7

Example 2: Solve 5x = 35.

示例 2：解方程 5x = 35。

5x ÷ 5 = 35 ÷ 5 => x = 7

Always use the inverse operation: addition reverses subtraction, multiplication reverses division, and vice versa.

始终使用逆运算：加法抵消减法，乘法抵消除法，反之亦然。

4. Solving Two-Step Equations | 解两步方程

Two-step equations require two inverse operations, usually in reverse order of BIDMAS. For example, in 3x – 7 = 8, x is first multiplied by 3 and then 7 is subtracted. To solve, undo the subtraction first, then undo the multiplication.

两步方程需要使用两次逆运算，通常按 BIDMAS 的相反顺序进行。例如在 3x – 7 = 8 中，x 先乘以 3，再减去 7。解方程时，先消除减法，再消除乘法。

Solve 3x – 7 = 8.

解方程 3x – 7 = 8。

3x – 7 + 7 = 8 + 7 => 3x = 15

3x ÷ 3 = 15 ÷ 3 => x = 5

It helps to write each new line directly below the previous one. This makes your working clear and easy to check in an exam.

把每一个新的步骤直接写在前一行下方会很有帮助。这样在考试中你的解题过程清晰且易于检查。

5. Equations with Brackets | 带括号的方程

When an equation contains brackets, expand them first using the distributive law. For example, 2(x + 4) = 18 becomes 2x + 8 = 18, then solve as a two-step equation.

当方程中含有括号时，首先使用分配律展开括号。例如 2(x + 4) = 18 可以展开为 2x + 8 = 18，然后按两步方程来解。

Solve 2(x + 4) = 18.

解方程 2(x + 4) = 18。

2x + 8 = 18

2x + 8 – 8 = 18 – 8 => 2x = 10

x = 5

Be careful with negative signs: 3(2x – 5) = 21 expands to 6x – 15 = 21, not 6x – 5 = 21.

注意负号：3(2x – 5) = 21 展开后是 6x – 15 = 21，而不是 6x – 5 = 21。

6. Equations with Variables on Both Sides | 两边都有未知数的方程

When the unknown appears on both sides, collect all variable terms on one side and all number terms on the other. Choose the side with the larger coefficient of x to keep things positive if possible.

当未知数出现在等式两边时，把所有含未知数的项移到一边，把所有常数项移到另一边。如果可能，选择 x 系数较大的一边，以保持未知数系数为正。

Solve 7x – 3 = 4x + 9.

解方程 7x – 3 = 4x + 9。

7x – 4x – 3 = 4x – 4x + 9 => 3x – 3 = 9

3x – 3 + 3 = 9 + 3 => 3x = 12

x = 4

Always subtract the smaller variable term from both sides to avoid negative coefficients at the start.

总是从两边减去较小的含未知数项，以避免一开始就出现负系数。

7. Forming Equations from Word Problems | 根据应用题建立方程

Word problems ask you to translate a real-life situation into an equation. Start by letting the unknown be x, then write each part of the sentence as a mathematical expression.

应用题要求你把现实情境转化为方程。首先设未知数为 x，然后把句子中的每一部分写成数学表达式。

Example: Sam is 5 years older than Tom. The sum of their ages is 27. Find Tom’s age.

示例：Sam 比 Tom 大 5 岁。他们的年龄之和是 27。求 Tom 的年龄。

Let Tom’s age = x. Then Sam’s age = x + 5.

设 Tom 的年龄为 x。则 Sam 的年龄为 x + 5。

x + (x + 5) = 27 => 2x + 5 = 27

2x = 22 => x = 11

Tom is 11 years old and Sam is 16 years old. Always re-read the question to check your answer makes sense in the context.

Tom 11 岁，Sam 16 岁。始终重新读一遍题目，检查答案在情境中是否合理。

8. Checking Your Solution | 检验你的解

After solving an equation, substitute your answer back into the original equation. If the left side equals the right side, your solution is correct.

解完方程后，把答案代回原方程。如果左边等于右边，你的解就是正确的。

For 3x – 7 = 8, we found x = 5.

对于 3x – 7 = 8，我们求出 x = 5。

Left side: 3(5) – 7 = 15 – 7 = 8 => Right side: 8

左边：3(5) – 7 = 15 – 7 = 8 => 右边：8

Since both sides equal 8, x = 5 is correct. This habit is especially useful in exams where small arithmetic slips are common.

由于两边都等于 8，x = 5 是正确的。这个习惯在考试中特别有用，因为小的算术失误很常见。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Many marks are lost in KS3 equations through avoidable errors. The table below shows some frequent mistakes and the correct approach.

在 KS3 方程题中，许多丢分都来自可以避免的错误。下表列出了一些常见错误和正确做法。

Mistake	Why it is wrong	Correct method
Forgetting to operate on both sides	The equation becomes unbalanced	Apply every operation to both sides
Adding instead of subtracting or vice versa	Uses the wrong inverse operation	Check what operation has been applied to x
Expanding brackets incorrectly	Misses multiplying all terms inside	Multiply every term inside the bracket
Losing a negative sign	Changes the value of the term	Keep the sign with the term when moving

错误	为什么错	正确方法
忘记在两边同时运算	等式不再平衡	对两边施加相同运算
该加却减，或该减却加	使用了错误的逆运算	检查对 x 施加了什么运算
去括号时出错	漏乘了括号内的某些项	把括号内每一项都相乘
丢掉了负号	改变了项的值	移项时保持符号跟随

10. Practice Questions and Quick Answers | 练习题与快速答案

Try these questions without looking at the answers. Then check your working step by step.

先尝试不看答案完成以下题目，然后逐步检查你的解题过程。

Solve x + 9 = 20
Solve 4x = 28
Solve 2x + 3 = 17
Solve 3(x – 2) = 12
Solve 5x + 4 = 2x + 19
Form and solve: Twice a number plus 6 equals 30. Find the number.

解方程 x + 9 = 20
解方程 4x = 28
解方程 2x + 3 = 17
解方程 3(x – 2) = 12
解方程 5x + 4 = 2x + 19
列方程求解：一个数的两倍加上 6 等于 30。求这个数。

Quick answers: x = 11; x = 7; x = 7; x = 6; x = 5; x = 12.

快速答案：x = 11；x = 7；x = 7；x = 6；x = 5；x = 12。

If you got any wrong, identify the step where the mistake happened and re-read the relevant section above.

如果你有任何一题做错，找出错误发生在哪一步，并重新阅读上文相关小节。

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Mastering Linear Equations for KS3 Cambridge Mathematics | KS3 剑桥数学：掌握线性方程