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 frequently in IGCSE Mathematics. This guide will help you master the techniques needed to solve them confidently, including equations with fractions, brackets, and variables on both sides.

线性方程是代数的基础，在IGCSE数学中频繁出现。本指南将帮助你掌握解题技巧，自信应对含有分数、括号以及变量出现在两边的方程。

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. For example, 3x + 5 = 20 is linear because x is raised to the power of 1 only.

线性方程是变量最高次数为1的方程。例如，3x + 5 = 20 是线性的，因为 x 的指数仅为1。

Linear equations can be written in many forms, but they always have one solution (unless they are identities or contradictions).

线性方程可以写成多种形式，但通常只有一个解（除非是恒等式或矛盾式）。

Example | 例子: 2x − 7 = 9
Solution | 解: Add 7 to both sides → 2x = 16 → x = 8

2x − 7 = 9 ⇒ 2x = 16 ⇒ x = 8

2. The Balance Method | 平衡法

The balance method means performing the same operation on both sides of the equation. Imagine a scale: whatever you do to one side, you must do to the other to keep it balanced.

平衡法是指对方程两边执行相同操作。想象一座天平：对一边做什么，就必须对另一边做什么，以保持平衡。

To isolate the variable, use inverse operations: addition and subtraction are inverses; multiplication and division are inverses.

要隔离变量，使用逆运算：加法与减法互为逆运算；乘法与除法互为逆运算。

Operation \| 操作	Inverse \| 逆运算
+ a	− a
− a	+ a
× a	÷ a (a ≠ 0)
÷ a	× a (a ≠ 0)

3. Solving ax + b = c | 解形如 ax + b = c 的方程

This is the simplest type. First remove the constant term from the variable side, then divide by the coefficient of x.

这是最简单的一类。首先从变量一侧去掉常数项，然后除以 x 的系数。

Worked Example | 例题: Solve 5x + 3 = 18

5x + 3 = 18 ⇒ 5x = 15 ⇒ x = 3

Check: 5(3) + 3 = 15 + 3 = 18 ✔

检验：5(3) + 3 = 15 + 3 = 18 ✔

Always substitute your answer back into the original equation to verify it.

始终将答案代入原方程进行检验。

4. Equations with Brackets | 含括号的方程

When brackets appear, expand them first using the distributive law: a(b + c) = ab + ac. Then solve as usual.

当出现括号时，先用分配律展开：a(b + c) = ab + ac。然后照常求解。

Worked Example | 例题: Solve 3(2x − 4) = 18

3(2x − 4) = 18 ⇒ 6x − 12 = 18 ⇒ 6x = 30 ⇒ x = 5

Be careful with negative signs outside brackets: −2(x − 3) = −2x + 6.

注意括号外的负号：−2(x − 3) = −2x + 6。

Expand | 展开
Collect like terms | 合并同类项
Isolate the variable | 分离变量

5. Variables on Both Sides | 变量出现在两边

When x appears on both sides, collect all x terms on one side and constants on the other. Choose the side that gives a positive coefficient for x to simplify.

当 x 出现在方程两边时，将所有含 x 的项集中到一边，常数集中到另一边。选择使 x 系数为正的一边可简化计算。

Worked Example | 例题: Solve 7x − 2 = 3x + 10

7x − 2 = 3x + 10 ⇒ 7x − 3x = 10 + 2 ⇒ 4x = 12 ⇒ x = 3

In general, you can move a term by adding its opposite to both sides.

一般来说，可以通过在两边同时加上某项的相反数来移项。

6. Equations with Fractions | 含分数的方程

To eliminate fractions, multiply every term by the lowest common denominator (LCD). This makes the equation easier to solve.

为消去分数，将每一项乘以最小公分母（LCD）。这会使方程更容易求解。

Worked Example | 例题: Solve x/2 + 1 = x/3 + 4

The LCD of 2 and 3 is 6. Multiply all terms by 6:

2和3的最小公分母是6。所有项乘以6：

6 × (x/2) + 6 × 1 = 6 × (x/3) + 6 × 4

3x + 6 = 2x + 24 ⇒ x = 18

Check: 18/2 + 1 = 9 + 1 = 10; 18/3 + 4 = 6 + 4 = 10 ✔

检验：18/2 + 1 = 9 + 1 = 10；18/3 + 4 = 6 + 4 = 10 ✔

7. Equations with the Variable in the Denominator | 变量在分母中的方程

If the variable appears in a denominator, multiply by that denominator to remove it. Be careful: the denominator cannot be zero.

如果变量出现在分母中，乘以该分母以去掉它。注意：分母不能为零。

Worked Example | 例题: Solve 5/(x − 2) = 3

5/(x − 2) = 3 ⇒ 5 = 3(x − 2) ⇒ 5 = 3x − 6 ⇒ 11 = 3x ⇒ x = 11/3

Always state the restriction: x ≠ 2 in this case.

始终注明限制条件：此题中 x ≠ 2。

8. Forming Linear Equations from Word Problems | 从应用题构建线性方程

Many exam questions require you to translate words into an equation. Define your variable first, then write relationships using that variable.

许多考试题目要求你将文字转化为方程。首先定义变量，然后用该变量写出关系式。

Example | 例子: The sum of a number and 9 is three times the number. Find the number.

例题：一个数与9的和等于这个数的3倍。求这个数。

Let the number be n. Then n + 9 = 3n.

设这个数为 n。则 n + 9 = 3n。

n + 9 = 3n ⇒ 9 = 2n ⇒ n = 4.5

Check: 4.5 + 9 = 13.5; 3 × 4.5 = 13.5 ✔

检验：4.5 + 9 = 13.5；3 × 4.5 = 13.5 ✔

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

Students often make errors when moving terms, expanding brackets, or forgetting to multiply every term by the same factor.

学生在移项、展开括号或忘记每一项都乘以同一因子时经常出错。

Mistake | 错误: 5x + 3 = 18 → 5x = 18 − 3 (correct) but writing 5x = 18 + 3 is wrong. | 5x = 15 正确，但写成 5x = 21 错误。
Mistake | 错误: −2(x − 3) = −2x − 3, but the correct expansion is −2x + 6. | 正确展开为 −2x + 6。
Mistake | 错误: Multiplying only one side by a number. Always multiply every term. | 只乘一边。必须每一项都乘。

Always write each step neatly and check your answer at the end.

每一步都要写清楚，最后检查答案。

10. Quick Exam Tips | 快速考试技巧

In IGCSE exams, show your working clearly to earn method marks even if you make a small arithmetic slip.

在IGCSE考试中，清晰展示步骤可以获得方法分，即使有小的计算失误。

Use the layout below for any linear equation:

对于任何线性方程，采用下面格式：

Original equation | 原方程

Step 1: expand / remove fractions | 第一步：展开/去分母

Step 2: collect like terms | 第二步：合并同类项

Step 3: isolate x | 第三步：分离 x

When your solution is a fraction, it is acceptable to leave it as an improper fraction unless the question asks for a decimal.

当解为分数时，除非题目要求小数，否则可保留为假分数。

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Solving Linear Equations | 解线性方程