Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are the foundation of algebra. In IGCSE Mathematics, you will meet them again and again, from simple one-step problems to multi-step equations with brackets and fractions. This article will guide you through the methods you need, step by step, with clear explanations and worked examples.

一元一次方程是代数的基石。在 IGCSE 数学中，你会反复遇到它们，从简单的一步运算到含括号和分数的多步方程。本文将一步一步带你掌握所需的方法，用清晰的讲解和完整的例题帮助你巩固知识。

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

A linear equation is an equation where the unknown (usually x) has an exponent of 1. It can be written in the form ax + b = c, where a, b and c are known numbers and a ≠ 0. The graph of a linear equation is a straight line, which is why we call it “linear”.

线性方程是指未知数（通常是 x）的次数为 1 的方程。它可以写成 ax + b = c 的形式，其中 a、b、c 是已知数，且 a ≠ 0。线性方程的图像是一条直线，因此得名 “linear”（线性的）。

For example: 3x + 5 = 14 is a linear equation. The unknown x appears only once and is not raised to a power such as x² or √x.

例如：3x + 5 = 14 就是一个线性方程。未知数 x 只出现一次，并且没有 x² 或 √x 这样的幂次或根式。

2. The Balance Method | 平衡法

Think of an equation like a balance scale. The equals sign means both sides have equal value. Whatever you do to one side, you must do to the other side to keep the scale balanced. This is the golden rule: preserve equality.

把方程想象成一架天平。等号表示两边数值相等。你对一边做的操作，也必须对另一边做，才能保持天平平衡。这就是黄金法则：保持等式成立。

Add the same number to both sides | 两边加同一个数
Subtract the same number from both sides | 两边减同一个数
Multiply both sides by the same number | 两边乘同一个数
Divide both sides by the same number (not 0) | 两边除以同一个非零数

Using this method keeps every step mathematically correct. It also helps you check your working later.

使用这种方法，每一步都保持数学上的正确性，也方便你之后检查解题过程。

3. Solving One-Step Equations | 解一步方程

One-step equations require only one operation to isolate x. For example:

一步方程只需要进行一次运算就能求出 x。例如：

x + 7 = 12

Subtract 7 from both sides: x = 5.

两边减去 7：x = 5。

x − 3 = 10

Add 3 to both sides: x = 13.

两边加 3：x = 13。

4x = 20

Divide both sides by 4: x = 5.

两边除以 4：x = 5。

x ÷ 6 = 3

Multiply both sides by 6: x = 18.

两边乘以 6：x = 18。

Remember that the operation you apply must always be the inverse (opposite) of the one that is currently happening to x.

记住：你所进行的运算必须是当前 x 所受运算的逆向（相反）运算。

4. Solving Two-Step Equations | 解两步方程

Two-step equations require two operations to isolate x. The typical form is ax + b = c. For example:

两步方程需要两次运算才能求出 x，常见形式为 ax + b = c。例如：

2x + 3 = 11

First, subtract 3 from both sides: 2x = 8.

第一步，两边减去 3：2x = 8。

Then, divide both sides by 2: x = 4.

然后，两边除以 2：x = 4。

Always deal with addition/subtraction first, then multiplication/division. This is the reverse of the “order of operations” (BIDMAS).

通常先处理加法/减法，再处理乘法/除法。这与运算顺序（BIDMAS）恰好相反。

Another example: 5x − 4 = 16. Add 4 to both sides: 5x = 20. Then divide by 5: x = 4.

另一个例子：5x − 4 = 16。两边加 4：5x = 20。再除以 5：x = 4。

5. Equations with Variables on Both Sides | 含未知数在方程两边的解法

When x appears on both sides of the equation, your first goal is to collect all x terms on one side and all constant terms on the other. For example:

当方程两边都出现 x 时，我们的首要目标是将所有含 x 的项移到一边，所有常数项移到另一边。例如：

3x + 5 = 2x + 9

Subtract 2x from both sides: x + 5 = 9.

两边减去 2x：x + 5 = 9。

Then subtract 5 from both sides: x = 4.

再两边减去 5：x = 4。

Alternatively, you could subtract 3x first, but it is often easier to move the smaller x term to avoid negatives. Be careful with signs when moving terms.

你也可以先减去 3x，但通常移动较小的含 x 项更容易避免负数。移动项时请特别留意符号。

Example with negative outcomes:

含负数结果的例子：

5x − 3 = 2x + 6

Subtract 2x: 3x − 3 = 6.

两边减 2x：3x − 3 = 6。

Add 3: 3x = 9, so x = 3.

两边加 3：3x = 9，因此 x = 3。

6. Equations with Brackets | 含括号的方程

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

如果方程含有括号，先用分配律展开括号：a(b + c) = ab + ac，然后再按常规方法求解。

3(2x + 1) = 15

Expand: 6x + 3 = 15.

展开：6x + 3 = 15。

Subtract 3: 6x = 12. Then divide by 6: x = 2.

两边减 3：6x = 12，再除以 6：x = 2。

Sometimes brackets appear on both sides. Expand all brackets first, then collect like terms.

有时方程两边都有括号。先展开所有括号，再合并同类项。

2(x + 3) = 3(x − 1)

Expand: 2x + 6 = 3x − 3.

展开：2x + 6 = 3x − 3。

Collect x terms: 6 + 3 = 3x − 2x, so 9 = x, i.e. x = 9.

集中含 x 的项：6 + 3 = 3x − 2x，因此 9 = x，即 x = 9。

7. Equations with Fractions | 含分数的方程

Fractions can make equations look harder, but the strategy is simple: multiply both sides by the denominator (or the lowest common denominator) to eliminate fractions.

分数会让方程看起来更复杂，但策略很简单：两边同时乘以分母（或最小公分母）来去掉分数。

x/2 + 3 = 7

Multiply every term by 2: x + 6 = 14, so x = 8.

每一项都乘以 2：x + 6 = 14，因此 x = 8。

For multiple fractions, multiply by the LCM of all denominators.

有多个分数时，乘以所有分母的最小公倍数（LCM）。

x/3 + x/4 = 7

The LCM of 3 and 4 is 12. Multiply every term by 12:

3 和 4 的最小公倍数是 12。每一项都乘以 12：

4x + 3x = 84

So 7x = 84, x = 12.

因此 7x = 84，x = 12。

Be careful: multiply every term, including whole numbers, not just the fractions.

注意：每一项都要乘，包括整数项，不能只乘分数。

8. Word Problems | 应用题

Word problems test your ability to translate real-life situations into equations. Follow these steps:

应用题考验你将实际情境转化为方程的能力。请遵循以下步骤：

Read the problem carefully and identify the unknown. | 仔细读题并确定未知数。
Let x represent that unknown. | 用 x 表示未知数。
Write an equation based on the information given. | 根据所给信息列出方程。
Solve the equation. | 解方程。
Check your answer in the context. | 在情境中检查答案是否合理。

Example: “A number is doubled and then 5 is added, giving 19. Find the number.”

例子：“一个数先乘以 2，再加 5，结果是 19。求这个数。”

Let x be the number. Then 2x + 5 = 19. Subtract 5: 2x = 14. Divide by 2: x = 7.

设这个数为 x。则 2x + 5 = 19。两边减 5：2x = 14。再除以 2：x = 7。

Always include units in your final answer when the problem involves measurements.

当问题涉及实际度量时，最终答案务必带上单位。

9. Checking Solutions | 验证解

After solving, substitute your value back into the original equation. If the left-hand side equals the right-hand side, your solution is correct.

解出答案后，将值代回原方程。如果左边等于右边，说明你的解是正确的。

Example: For 3x + 5 = 14, we found x = 3. Check: 3 × 3 + 5 = 9 + 5 = 14. True.

例子：对于 3x + 5 = 14，我们解得 x = 3。验证：3 × 3 + 5 = 9 + 5 = 14，成立。

Checking is especially important in word problems where the answer must make sense. For example, if x represents the number of people, a negative value would be impossible.

验证在应用题中尤其重要，因为答案必须符合实际意义。例如，若 x 代表人的人数，负值当然不可能。

10. Common Mistakes | 常见错误

Many students lose marks for simple errors. The table below shows common mistakes and how to avoid them.

许多学生因为简单失误而丢分。下表列出常见错误及避免方法。

Mistake \| 错误	Correct Approach \| 正确做法
Forgetting to apply the same operation to both sides 只对一边进行运算	Always change both sides equally 始终对两边同时进行相同变化
Wrong sign when moving terms 移项时符号错误	When moving a term, reverse its sign 移项时要变号
Not multiplying every term by the LCM 没有将每一项都乘以最小公倍数	Multiply every single term, including integers 每一项都要乘，包括整数
Expanding brackets incorrectly 展开括号错误	Use a(b + c) = ab + ac carefully 仔细使用分配律 a(b + c) = ab + ac

Be especially careful with negative signs and fractions, as they cause the most errors.

要特别小心负号和分数，它们最容易导致错误。

11. Practice Questions | 练习题

Try these problems on your own. Answers are provided below.

请自己尝试以下题目，答案在下方。

3x − 7 = 11 | 3x − 7 = 11
5(x + 2) = 20 | 5(x + 2) = 20
2x + 3 = x + 8 | 2x + 3 = x + 8
x/3 + 2 = 6 | x/3 + 2 = 6
4(2x − 1) = 3(x + 2) | 4(2x − 1) = 3(x + 2)

Answers:

答案：

x = 6
x = 2
x = 5
x = 12
x = 2

If you got any wrong, review the relevant section before moving on.

如果有做错的，请先回顾相应章节再继续学习。

12. Summary | 总结

To solve any linear equation, remember the key steps:

解任何一元一次方程，请记住关键步骤：

Simplify both sides by removing brackets and collecting like terms. | 先去括号、合并同类项，化简两边。
Use inverse operations to isolate x. | 用逆运算孤立 x。
Keep the equation balanced by doing the same to both sides. | 两边做相同操作，保持等式平衡。
Check your answer by substitution. | 代回原方程验证答案。

Mastering linear equations gives you confidence for more advanced topics like simultaneous equations, inequalities and quadratic equations. Practice regularly, and you will soon solve them quickly and accurately.

掌握一元一次方程会为你学习更高级的专题建立信心，如联立方程、不等式和二次方程。坚持练习，你很快就能快速、准确地解出它们。

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Solving Linear Equations | 解一元一次方程