Solving Linear Equations Step by Step | 一元一次方程解法详解

📚 Solving Linear Equations Step by Step | 一元一次方程解法详解

Linear equations are one of the first truly algebraic tools you meet in KS3 Cambridge Mathematics. They turn a word problem or a balancing idea into a short mathematical sentence with an unknown value, usually written as x. Being able to solve an equation confidently is not just a checkpoint skill; it is the foundation for graphs, sequences, formulas, and many later topics in IGCSE and beyond.

一元一次方程是你在 KS3 剑桥数学中最早接触的真正代数工具之一。它把文字题或天平平衡的思想转化为一个含有未知数（通常写作 x）的简短数学语句。能够熟练解方程不仅是一项 checkpoint 技能，更是日后学习图像、数列、公式以及 IGCSE 乃至更高阶内容的基础。

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

A linear equation is an equation where the unknown appears only to the power of 1. The word ‘linear’ tells you that if you drew the relationship, it would form a straight line. Examples include 3x + 5 = 20, 7 = 2y – 1, and 4(x + 2) = 28. The key idea is that an equation states two expressions are equal, and our job is to find the value of the unknown that makes the statement true.

一元一次方程是指未知数只出现一次幂（即指数为 1）的方程。’线性’ 一词表示如果你把关系画出来，它会形成一条直线。例如 3x + 5 = 20，7 = 2y – 1 和 4(x + 2) = 28。核心思想是：方程说明两个表达式相等，而我们的任务是找出使这个等式成立的未知数的值。

2. The Balance Method | 天平法（等式平衡法）

Think of an equation as a set of old-fashioned balance scales. Whatever you do to one side, you must do exactly the same to the other side, or the balance is broken. This is the golden rule of algebra: keep the equation balanced. If you add 4 to the left side, add 4 to the right side. If you divide the left side by 3, divide the right side by 3 too.

把方程想象成一架老式天平。无论你对一边做什么操作，都必须对另一边做完全相同的操作，否则天平就会失衡。这是代数的黄金法则：保持方程平衡。如果你在左边加 4，也要在右边加 4。如果你把左边除以 3，右边也要除以 3。

Add or subtract the same number on both sides | 两边同时加或减同一个数
Multiply or divide both sides by the same non-zero number | 两边同时乘或除以同一个非零数
Expand brackets carefully before balancing | 先仔细展开括号，再保持平衡

These three rules are enough to solve almost every linear equation you will meet at KS3 level.

这三条规则足以解决你在 KS3 阶段遇到的几乎所有一元一次方程。

3. Solving by Using Inverse Operations | 利用逆运算求解

To solve a simple equation such as x + 7 = 15, we use inverse operations. Addition and subtraction are inverse operations; multiplication and division are inverse operations. Since x has 7 added to it, we subtract 7 from both sides:

要解像 x + 7 = 15 这样的简单方程，我们使用逆运算。加法和减法互为逆运算；乘法和除法互为逆运算。因为 x 被加上了 7，所以我们在两边同时减去 7：

x + 7 – 7 = 15 – 7

x = 8

For multiplication, if 3x = 21, we divide both sides by 3 because division undoes multiplication. This gives x = 7. Always use the opposite operation to free the unknown step by step.

对于乘法，如果 3x = 21，我们两边同时除以 3，因为除法能撤销乘法。这样得到 x = 7。始终使用相反运算一步步把未知数解出来。

4. Equations with Brackets | 含括号的方程

When an equation contains brackets, start by expanding them. For 3(x + 4) = 27, expand the left-hand side to get 3x + 12 = 27. Then subtract 12 from both sides to obtain 3x = 15, and finally divide both sides by 3 to get x = 5. Always expand before you start moving terms unless the bracket is being multiplied by a factor that you can divide away first.

当方程含有括号时，首先要展开括号。对于 3(x + 4) = 27，将左边展开得到 3x + 12 = 27。然后两边同时减去 12，得到 3x = 15，最后两边同时除以 3，得到 x = 5。除非括号前面的因数可以先除过去，否则通常先展开括号，再移项。

Be especially careful with negative signs when expanding. For 2(3x – 5) = 14, expand to get 6x – 10 = 14, then add 10 to both sides and divide by 6 to find x = 4.

展开时尤其要注意负号。对于 2(3x – 5) = 14，展开得到 6x – 10 = 14，然后两边加 10，再除以 6，求得 x = 4。

5. Equations with Unknowns on Both Sides | 未知数在等式两边的方程

Some equations have x on both sides, such as 5x – 3 = 2x + 9. The first goal is to collect all x terms on one side and all number terms on the other. Subtract 2x from both sides to get 3x – 3 = 9. Then add 3 to both sides to get 3x = 12. Divide by 3 to find x = 4.

有些方程的两边都有 x，例如 5x – 3 = 2x + 9。首要目标是把所有含 x 的项集中到一边，把所有数字项集中到另一边。两边同时减去 2x，得到 3x – 3 = 9。然后两边同时加 3，得到 3x = 12。再除以 3，求得 x = 4。

It is usually easier to keep the x term positive. If you end up with -2x = 8, multiply both sides by -1 to get 2x = -8, then divide by 2.

通常让 x 项保持为正会更方便。如果你最后得到 -2x = 8，可以把两边同时乘以 -1，得到 2x = -8，再除以 2。

6. Equations with Fractions | 含分数的方程

Equations with fractions look harder, but the trick is to clear the fractions first. For x/3 + 2 = 6, subtract 2 from both sides to get x/3 = 4, then multiply both sides by 3 to get x = 12. For an equation like (2x)/5 = 4, multiply both sides by 5 to get 2x = 20, then divide by 2.

含分数的方程看起来更难，但诀窍是先消去分母。对于 x/3 + 2 = 6，两边先减 2 得到 x/3 = 4，然后两边同时乘以 3，得到 x = 12。对于 (2x)/5 = 4 这样的方程，两边同乘 5，得到 2x = 20，再除以 2。

You can also multiply every term by the lowest common multiple of the denominators at the start. For x/2 + x/3 = 5, multiply each term by 6 to get 3x + 2x = 30, so 5x = 30 and x = 6.

你也可以一开始就用所有分母的最小公倍数去乘每一项。对于 x/2 + x/3 = 5，用 6 乘每一项，得到 3x + 2x = 30，因此 5x = 30，x = 6。

7. Checking Your Answer | 检验答案

After finding a value for x, always substitute it back into the original equation to check. If you solved 4(x – 2) = 16 and got x = 6, check: left-hand side = 4(6 – 2) = 4 × 4 = 16, which equals the right-hand side. If it does not match, go back and check each step. Checking is quick and can catch most errors in signs, brackets, and arithmetic.

求出 x 的值后，一定要把它代回原方程检验。如果你解 4(x – 2) = 16 得到 x = 6，就检验：左边 = 4(6 – 2) = 4 × 4 = 16，与右边相等。如果两边不相等，就返回去逐步检查。检验很快，而且能发现大部分符号、括号和计算错误。

Always substitute into the original equation, not the simplified one, because the original equation is the statement you were asked to solve.

一定要代回原方程，而不是化简后的方程，因为原方程才是题目要求你解的等式。

8. Common Mistakes to Avoid | 常见错误与避免方法

Forgetting to do the same operation on both sides | 忘记两边做相同的运算
Mixing up inverse operations, e.g. subtracting instead of dividing | 混淆逆运算，例如该除以却减去
Expanding brackets incorrectly, especially with negative signs | 展开括号出错，尤其是负号情况
Losing the negative sign when moving terms across | 移项时丢掉负号
Not checking the final answer | 不检验最终答案

Many errors come from rushing. Write each line clearly, keep the equals signs aligned, and show every step until the process becomes automatic.

很多错误来自匆忙。每一行都要写得清楚，保持等号对齐，并写出每一步，直到整个过程变得熟练自然。

9. Word Problems Leading to Equations | 由实际问题列方程

To turn a word problem into an equation, first choose a letter for the unknown. For example: ‘I think of a number, multiply it by 5, add 8, and get 33.’ Let the number be n. Then the statement becomes 5n + 8 = 33. Subtract 8 from both sides to get 5n = 25, then divide by 5 to get n = 5.

要把文字题转化为方程，首先用一个字母表示未知数。例如：’我想一个数，把它乘以 5，再加 8，得到 33。’ 设这个数为 n。那么这句话就变成 5n + 8 = 33。两边减 8 得到 5n = 25，再除以 5 得到 n = 5。

Translate words carefully: ‘more than’ often means addition, ‘less than’ means subtraction, ‘times’ means multiplication, and ‘shared equally’ means division. Always define your letter at the start.

翻译文字时要仔细：’比……多’ 通常表示加法，’比……少’ 表示减法，’……倍’ 表示乘法，’平均分’ 表示除法。开始时一定要定义你设的字母代表什么。

10. Practice Questions and Answers | 练习题与答案

Try these questions before looking at the answers. Solve each equation and check your answer.

在查看答案之前先尝试做以下题目。解出每个方程并检验答案。

Question \| 题目	Answer \| 答案
x + 9 = 14	x = 5
7x – 2 = 26	x = 4
3(2x – 1) = 27	x = 5
4x + 6 = 2x + 18	x = 6
x/4 + 1 = 5	x = 16
(3x)/2 – 1 = 8	x = 6

If you got any of these wrong, go back and find the exact line where your solution stopped satisfying the equality. That is where the error happened.

如果这些题目中有任何一题做错，就回去找到你的解题过程中不再满足等式的那一行。错误就发生在那里。

11. Summary | 总结

To solve a linear equation: expand brackets if needed, collect all unknown terms on one side and numbers on the other using inverse operations, simplify, divide to find the unknown, and always check by substitution. The balance method works for every linear equation at KS3 level.

解一元一次方程的步骤是：必要时展开括号，用逆运算把所有未知项移到一边、数字项移到另一边，化简，再除以未知数的系数求出未知数，最后一定要代入检验。天平法适用于 KS3 阶段的每一类一元一次方程。

Mastering this topic gives you confidence for harder algebra in Cambridge Lower Secondary and beyond. Practice slowly first, then increase speed while keeping every step visible.

掌握这个主题会让你在剑桥初中及更高阶段学习更难的代数时更有信心。先慢慢练习，然后提高速度，同时保持每一步清晰可见。

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Solving Linear Equations Step by Step | 一元一次方程解法详解