Solving Linear Equations | 解线性方程

📚 Solving Linear Equations | 解线性方程

Linear equations are the foundation of algebra and appear in every area of A-Level Mathematics. Mastering them ensures accuracy in solving simultaneous equations, inequalities, and even calculus problems. This revision guide breaks down the essential methods step by step.

线性方程是代数的基石，出现在A-Level数学的每一个领域中。掌握它们能确保你在解联立方程、不等式甚至微积分问题时保持准确性。本复习指南将逐步拆解核心方法。

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

A linear equation is an equation where the highest power of the variable is 1. It can be written in the form ax + b = c, where a, b, and c are constants and a ≠ 0. The graph of a linear equation is a straight line.

线性方程是变量最高次数为1的方程。它可以写成 ax + b = c 的形式，其中 a、b、c 为常数，且 a ≠ 0。线性方程的图像是一条直线。

In A-Level Mathematics, you may encounter linear equations with one variable, two variables, or even parameters. The goal is always the same: isolate the variable.

在A-Level数学中，你可能会遇到含一个变量、两个变量甚至参数的线性方程。目标始终相同：分离变量。

2. The Balance Principle | 等式平衡原则

An equation behaves like a balance scale. Whatever you do to one side, you must do to the other. This preserves equality and allows you to rearrange terms safely.

等式就像一架天平。你对一边做的任何操作，都必须在另一边做相同操作。这样才能保持等号成立，并允许你安全地重新排列各项。

‘Add or subtract the same quantity on both sides.’

两边同时加或减同一个量。
‘Multiply or divide both sides by the same non-zero quantity.’

两边同时乘或除以同一个非零量。

If x + 3 = 10, then x + 3 − 3 = 10 − 3, so x = 7.

若 x + 3 = 10，则 x + 3 − 3 = 10 − 3，所以 x = 7。

3. Solving One-Step Equations | 解一步方程

One-step equations require only a single inverse operation. Identify the operation applied to the variable, then apply the opposite operation to both sides.

一步方程只需要一个逆运算。识别施加在变量上的运算，然后对两边执行相反的运算。

Example: Solve x/4 = 5.

例：解 x/4 = 5。

Since x is divided by 4, multiply both sides by 4:

由于 x 被除以 4，两边乘以 4：

4 × (x/4) = 4 × 5 → x = 20

Always state the value clearly and check by substitution.

务必清晰写出解的值，并通过代入检验。

4. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations, usually addition/subtraction and multiplication/division. The order of inverse operations matters: first undo addition/subtraction, then undo multiplication/division.

两步方程涉及两种运算，通常是加减和乘除。逆运算的顺序很重要：先消去加减，再消去乘除。

Example: Solve 3x + 4 = 19.

例：解 3x + 4 = 19。

Subtract 4 from both sides: 3x = 15

两边减 4：3x = 15
Divide both sides by 3: x = 5

两边除以 3：x = 5

Check: 3(5) + 4 = 19 ✓

检验：3(5) + 4 = 19 ✓

5. Equations with Brackets | 含括号的方程

When brackets appear, expand them first using the distributive law: a(b + c) = ab + ac. After expanding, collect like terms before isolating the variable.

当出现括号时，先用分配律展开括号：a(b + c) = ab + ac。展开后先合并同类项，再分离变量。

Example: Solve 2(3x − 1) = 4x + 10.

例：解 2(3x − 1) = 4x + 10。

Expand: 6x − 2 = 4x + 10

展开：6x − 2 = 4x + 10

Subtract 4x from both sides: 2x − 2 = 10

两边减 4x：2x − 2 = 10

Add 2 to both sides: 2x = 12, so x = 6.

两边加 2：2x = 12，所以 x = 6。

6. Equations with Fractions | 含分数的方程

Fractions can be eliminated by multiplying both sides of the equation by the common denominator of all denominators. This simplifies the equation into an easier integer form.

可以通过将方程两边乘以所有分母的公分母来消去分数。这能将方程简化为更易处理的整数形式。

Example: Solve (x/2) + (x/3) = 5.

例：解 (x/2) + (x/3) = 5。

The common denominator of 2 and 3 is 6. Multiply every term by 6:

2 和 3 的公分母是 6。每一项都乘以 6：

6 × (x/2) + 6 × (x/3) = 6 × 5 → 3x + 2x = 30 → 5x = 30 → x = 6

Be careful: when multiplying a term like (x + 1)/2 by the denominator, multiply the entire numerator: 6 × (x + 1)/2 = 3(x + 1).

注意：例如将 (x + 1)/2 乘以分母时，必须乘整个分子：6 × (x + 1)/2 = 3(x + 1)。

7. Variable on Both Sides | 变量出现在等号两边

Collect all variable terms on one side and constant terms on the other. The side you choose does not matter, but it is often easier to keep the coefficient of x positive.

将所有含变量的项移到等号一边，常数项移到另一边。选择哪一边不影响结果，但通常让 x 的系数为正会更方便。

Example: Solve 5x − 3 = 2x + 9.

例：解 5x − 3 = 2x + 9。

Subtract 2x from both sides: 3x − 3 = 9

两边减 2x：3x − 3 = 9

Add 3 to both sides: 3x = 12, so x = 4.

两边加 3：3x = 12，所以 x = 4。

8. Equations with Decimals and Percentages | 含小数和百分数的方程

Decimals and percentages can be converted into fractions or integers. For instance, 0.25 = 1/4 and 25% = 1/4. Multiply by a power of 10 to clear decimals, or write percentages as fractions first.

小数和百分数可以转换成分数或整数。例如，0.25 = 1/4，25% = 1/4。通过乘以 10 的幂消去小数，或者先把百分数写成分数。

Example: Solve 0.4x + 1.5 = 2.3.

例：解 0.4x + 1.5 = 2.3。

Multiply both sides by 10 to clear decimals: 4x + 15 = 23

两边乘以 10 消去小数：4x + 15 = 23

Then 4x = 8, so x = 2.

于是 4x = 8，所以 x = 2。

9. Constructing Linear Equations from Word Problems | 由应用题构建线性方程

Word problems require translating situations into algebraic language. Define a variable, identify relationships, and write an equation that models the problem.

应用题需要将情境转化为代数语言。先定义变量，识别数量关系，然后写出能建模问题的方程。

Example: A rectangle has length 4 cm more than twice its width. Its perimeter is 32 cm. Find the width.

例：一个长方形的长比宽的2倍多4 cm，周长为32 cm。求宽。

Let w be the width. Then length = 2w + 4.

设宽为 w。则长为 2w + 4。

Perimeter: 2(w + (2w + 4)) = 32

周长：2(w + (2w + 4)) = 32

Simplify: 2(3w + 4) = 32 → 6w + 8 = 32 → w = 4.

化简：2(3w + 4) = 32 → 6w + 8 = 32 → w = 4。

10. Special Cases: No Solution and Identities | 特殊情况：无解与恒等式

Not every linear equation has a unique solution. There are two special cases you must recognise:

并非所有线性方程都有唯一解。你必须识别两种特殊情况：

No solution: The equation simplifies to a false statement, e.g. 2x + 3 = 2x + 5 gives 3 = 5. Such an equation has no solution.

无解：方程化简为假命题，例如 2x + 3 = 2x + 5 得到 3 = 5。这样的方程无解。
Identity: The equation simplifies to a true statement, e.g. 4x + 6 = 2(2x + 3) gives 4x + 6 = 4x + 6. This is true for all x.

恒等式：方程化简为真命题，例如 4x + 6 = 2(2x + 3) 得到 4x + 6 = 4x + 6。它对所有 x 都成立。

When solving, if you eliminate x and get 0 = 0, the solution is all real numbers. If you get 0 = k (k ≠ 0), there is no solution.

在求解过程中，如果消去 x 后得到 0 = 0，则解为全体实数。如果得到 0 = k（k ≠ 0），则无解。

11. Checking Your Answer | 检查答案

Always substitute your solution back into the original equation. This verifies correctness and catches arithmetic mistakes.

务必把解代入原方程进行检验。这能验证正确性并发现算术错误。

For example, for 2x − 7 = 1, if you get x = 4, then 2(4) − 7 = 1 ✓.

例如，对于 2x − 7 = 1，如果你得到 x = 4，则 2(4) − 7 = 1 ✓。

Check that the value satisfies any given conditions, such as positive integer or being within a specified domain.

还要检查该解是否满足题目给出的条件，例如是否为正整数或在指定范围内。

12. Common Mistakes to Avoid | 常见错误提醒

Mistake / 错误	Correct method / 正确方法
Forgetting to multiply every term when clearing fractions / 去分母时漏乘某些项	Multiply every term by the common denominator / 每一项都乘以公分母
Sign errors when moving terms across the equals sign / 移项时符号错误	Add or subtract the same value from both sides / 两边做相同的加减运算
Not expanding brackets correctly / 括号展开错误	Use distributive law and be careful with negative signs / 使用分配律并注意负号
Dividing by zero when coefficient is zero / 当系数为0时除以0	Recognise special cases before dividing / 在除法前先识别特殊情况

By avoiding these pitfalls and practising systematically, you will solve linear equations accurately and confidently in any A-Level exam.

通过避免这些陷阱并系统练习，你将能在任何A-Level考试中准确、自信地求解线性方程。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Solving Linear Equations | 解线性方程