Mastering Linear Equations for KS3 Maths | 掌握 KS3 数学中的一元一次方程

📚 Mastering Linear Equations for KS3 Maths | 掌握 KS3 数学中的一元一次方程

Linear equations are the foundation of algebra at KS3. In the Cambridge Lower Secondary Mathematics curriculum, you are expected to set up, solve and check one-variable equations with confidence. This article breaks down the key methods, common pitfalls and exam-style practice you need for success.

一元一次方程是 KS3 代数的基础。在剑桥初中数学课程中，你需要能够熟练地建立、求解并检验一元方程。本文拆解关键方法、常见错误和考试题型，帮助你在这一考点上取得成功。

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1. What Is a Linear Equation? | 什么是一元一次方程？

A linear equation is an equation in which the unknown, usually written as x, is only raised to the power of 1. It can be written in the form ax + b = c, where a, b and c are known numbers.

一元一次方程是指未知数（通常写作 x）的指数仅为 1 的方程。它可以写成 ax + b = c 的形式，其中 a、b 和 c 是已知数。

For example, 2x + 3 = 11 is linear because x appears without any square, cube or root. Equations like x² = 9 or 1/x = 4 are not linear at KS3 level.

例如，2x + 3 = 11 是一元一次方程，因为 x 没有平方、立方或根号。像 x² = 9 或 1/x = 4 这样的方程在 KS3 阶段不属于一元一次方程。

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2. The Balance Method | 天秤法

The balance method is the key idea behind solving linear equations. Think of an equation as a balanced set of scales: whatever you do to one side, you must do to the other side to keep it balanced.

天秤法是解一元一次方程的核心思想。把方程想象成一架平衡的天平：你对一边做什么操作，必须对另一边做同样的操作，才能保持平衡。

If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side. This rule keeps the equation true while you isolate the unknown.

如果你对一边加、减、乘或除以一个数，你必须对另一边进行完全相同的操作。这一规则在分离未知数的过程中保持方程成立。

• Add the same value to both sides | 两边同时加上同一个数
• Subtract the same value from both sides | 两边同时减去同一个数
• Multiply or divide both sides by the same non-zero value | 两边同时乘以或除以同一个非零数

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3. Solving One-Step Equations | 解一步方程

A one-step equation needs only one inverse operation to solve. The inverse of addition is subtraction, and the inverse of multiplication is division.

一步方程只需要一次逆运算即可求解。加法的逆运算是减法，乘法的逆运算是除法。

To solve x + 7 = 15, subtract 7 from both sides:

要解 x + 7 = 15，两边同时减去 7：

x + 7 − 7 = 15 − 7

x = 8

To solve 4x = 28, divide both sides by 4:

要解 4x = 28，两边同时除以 4：

4x ÷ 4 = 28 ÷ 4

x = 7

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4. Solving Two-Step Equations | 解两步方程

A two-step equation combines two operations, such as 2x + 5 = 17. The golden rule is to undo addition or subtraction first, then undo multiplication or division.

两步方程包含两种运算，例如 2x + 5 = 17。黄金法则是先消去加法或减法，再消去乘法或除法。

Solve 2x + 5 = 17:

解 2x + 5 = 17：

2x + 5 − 5 = 17 − 5

2x = 12

2x ÷ 2 = 12 ÷ 2

x = 6

Another common form is 3x − 4 = 20. Add 4 to both sides first, then divide by 3:

另一种常见形式是 3x − 4 = 20。先两边加上 4，再除以 3：

3x − 4 + 4 = 20 + 4

3x = 24

x = 8

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5. Expanding Brackets First | 先去括号

When a linear equation contains brackets, expand them before using the balance method. This rule follows the order of operations in reverse.

当一元一次方程含有括号时，先展开括号，再使用天秤法。这一规则是逆用运算顺序。

Solve 3(x + 2) = 21:

解 3(x + 2) = 21：

3x + 6 = 21

3x = 15

x = 5

You can also solve by dividing both sides by the coefficient of the bracket first: 3(x + 2) ÷ 3 = 21 ÷ 3, giving x + 2 = 7, then x = 5. Both methods are valid.

你也可以先两边除以括号前的系数：3(x + 2) ÷ 3 = 21 ÷ 3，得到 x + 2 = 7，再得 x = 5。两种方法都正确。

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6. Equations with Variables on Both Sides | 两边含未知数的方程

Some equations have the unknown on both sides, such as 4x + 3 = 2x + 11. Start by collecting the x terms on one side and the numbers on the other.

有些方程两边都含有未知数，例如 4x + 3 = 2x + 11。先把含 x 的项移到一边，把数字移到另一边。

Subtract 2x from both sides:

两边同时减去 2x：

4x − 2x + 3 = 2x − 2x + 11

2x + 3 = 11

Then subtract 3 and divide by 2:

然后减去 3，再除以 2：

2x = 8

x = 4

Remember that you can move terms to either side as long as you apply the inverse operation correctly. Choose the side that keeps the coefficient positive when possible.

请记住，只要正确使用逆运算，你可以把项移到任意一边。尽量选择使系数为正的一边。

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7. Equations with Fractions | 含分数的方程

To solve an equation like x/3 + 2 = 7, first subtract 2 from both sides, then multiply by the denominator.

要解像 x/3 + 2 = 7 这样的方程，先两边减去 2，再乘以分母。

x/3 + 2 − 2 = 7 − 2

x/3 = 5

x/3 × 3 = 5 × 3

x = 15

For equations like (2x + 1)/3 = 5, multiply both sides by the denominator first, then solve the two-step equation:

对于像 (2x + 1)/3 = 5 这样的方程，先两边乘以分母，再解两步方程：

2x + 1 = 15

2x = 14

x = 7

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8. Writing Equations from Word Problems | 根据文字题建立方程

Many Cambridge exam questions ask you to form an equation from a real-life situation. Read the problem carefully and translate each phrase into algebra.

许多剑桥考试题要求你从实际情境中建立方程。仔细阅读题目，把每个短语转化为代数语言。

• “Three times a number” means 3x | “一个数的三倍” 表示 3x
• “Five more than a number” means x + 5 | “比一个数多五” 表示 x + 5
• “The sum of a number and 8 is 20” means x + 8 = 20 | “一个数与 8 的和是 20” 表示 x + 8 = 20

Example: I think of a number, multiply it by 4, then add 7. The answer is 31. Let the number be x:

例子：我想一个数，乘以 4，再加 7，结果是 31。设这个数为 x：

4x + 7 = 31

4x = 24

x = 6

Always define your variable clearly before forming the equation, then check that the answer makes sense in the original context.

在建立方程前，始终清晰地设出变量，然后检查答案在原始语境中是否合理。

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9. Checking Your Solution | 检验你的解

Always substitute your final x value back into the original equation to verify it works. This habit catches careless mistakes and shows full working in exams.

始终把你的最终 x 值代回原方程进行检验。这个习惯可以避免粗心错误，并在考试中展示完整步骤。

For 2x + 5 = 17, if you found x = 6, check:

对于 2x + 5 = 17，如果你得到 x = 6，检验：

2 × 6 + 5 = 12 + 5 = 17

Since the left side equals the right side, the solution is correct.

因为左边等于右边，所以解是正确的。

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10. Common Mistakes to Avoid | 常见错误

Many marks are lost through simple errors. The table below shows the most common mistakes and how to correct them.

很多分数因简单错误而丢失。下表列出最常见的错误及改正方法。

Mistake | 错误	Correct approach | 正确处理
Forgetting to operate on both sides | 忘记两边同时操作	Always apply the same operation to both sides | 始终对两边进行相同操作
Undoing operations in the wrong order | 逆运算顺序错误	Undo addition/subtraction before multiplication/division | 先逆加减，再逆乘除
Mis-handling negative signs | 负号处理错误	Use inverse operations carefully with negatives | 处理负数时谨慎使用逆运算
Expanding brackets incorrectly | 括号展开错误	Multiply every term inside the bracket | 乘以括号内的每一项

Pay particular attention to equations like 5x − 3 = 2x + 9. A common error is to subtract 2x from only one side or to subtract 3 incorrectly.

特别注意像 5x − 3 = 2x + 9 这样的方程。常见错误是只从一边减去 2x，或者错误地减去 3。

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11. Exam-Style Practice Questions | 考试题型练习

Try these questions to test your understanding. Solve each equation, showing every step.

尝试以下题目检验你的理解。解每个方程，展示每一步。

• Solve 5x + 2 = 37 | 解 5x + 2 = 37
• Solve 2(x − 3) = 14 | 解 2(x − 3) = 14
• Solve 3x + 4 = x + 16 | 解 3x + 4 = x + 16
• Solve x/4 − 1 = 6 | 解 x/4 − 1 = 6

Worked solutions:

参考解答：

For 5x + 2 = 37: subtract 2, then divide by 5, giving x = 7.

对于 5x + 2 = 37：先减 2，再除以 5，得到 x = 7。

For 2(x − 3) = 14: expand to 2x − 6 = 14, add 6, then divide by 2, giving x = 10.

对于 2(x − 3) = 14：展开得 2x − 6 = 14，加 6，再除以 2，得到 x = 10。

For 3x + 4 = x + 16: subtract x and subtract 4, giving 2x = 12, so x = 6.

对于 3x + 4 = x + 16：减去 x 和 4，得到 2x = 12，因此 x = 6。

For x/4 − 1 = 6: add 1, then multiply by 4, giving x = 28.

对于 x/4 − 1 = 6：加 1，再乘以 4，得到 x = 28。

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12. Summary and Key Tips | 总结与关键提示

To master linear equations, always use the balance method, undo operations in the correct order, and check your answer by substitution. Practice translating word problems into equations and be careful with negative numbers and brackets.

要掌握一元一次方程，始终使用天秤法，按正确顺序进行逆运算，并通过代入检验答案。练习将文字题转化为方程，并注意负数和括号。

Keep your working neat and write one operation per line. This makes it easier to spot mistakes and helps you gain method marks in Cambridge exams.

保持解题过程整洁，每行只写一步操作。这样更容易发现错误，并帮助你在剑桥考试中获得步骤分。

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