📚 Mastering Linear Equations: From First Steps to Multi-Step Problems | 掌握线性方程：从入门到多步解题
Linear equations are the foundation of algebra at KS3 and appear throughout the Cambridge Lower Secondary Mathematics curriculum. Being able to solve equations quickly and accurately is essential for later topics such as graphs, sequences, and real-life problem solving. In this article, we will build your confidence step by step, from simple one-step equations to equations with brackets, fractions, and variables on both sides.
线性方程是 KS3 阶段代数的基础，贯穿剑桥初中数学课程的各个部分。能够快速、准确地解方程，对后续学习图像、数列和现实问题解决等内容至关重要。本文将逐步建立你的信心，从简单的一步方程到含括号、分数以及两边都有变量的方程。
1. What Is a Linear Equation? | 什么是线性方程？
, or 1/x. For example, x + 3 = 7 is a linear equation because the highest power of x is 1.
线性方程是指未知数（通常写成 x）的指数仅为 1 的方程。这意味着方程中没有 x²、x³ 或 1/x 这样的项。例如，x + 3 = 7 是一个线性方程，因为 x 的最高次数是 1。
The word 'linear' comes from the fact that the graph of such an equation is a straight line. When you solve a linear equation, you are finding the value of x that makes the equation true. This value is called the solution or the root.
“线性”一词来源于这类方程的图像是一条直线。解线性方程时，你要求出使方程成立的 x 的值。这个值被称为方程的解或根。
At KS3, you will meet linear equations in many forms. Some have one step to solve, some have two or more, and some require you to expand brackets or collect like terms first. The key is to understand the balancing method, which we will look at next.
在 KS3 阶段，你会遇到各种形式的线性方程。有些只需要一步就能解出，有些需要两步或更多步，有些则需要先去括号或合并同类项。关键是要理解平衡法，接下来我们将学习这一方法。
2. Keeping the Balance: The Golden Rule | 保持平衡：黄金法则
An equation is like a balance scale. The left side and the right side are equal in value. If you add, subtract, multiply, or divide one side by a number, you must do exactly the same thing to the other side. This is often called the golden rule of equations.
方程就像一架天平。左边和右边的值相等。如果你对一边加、减、乘或除以一个数，就必须对另一边做完全相同的运算。这通常被称为方程的黄金法则。
r example, if x + 3 = 10, we can subtract 3 from both sides to keep the equation balanced:
例如，如果 x + 3 = 10，我们可以在两边同时减去 3 来保持方程平衡：
x + 3 − 3 = 10 − 3 → x = 7
This rule allows us to isolate x on one side of the equation. The opposite operation is always used: addition is undone by subtraction, subtraction by addition, multiplication by division, and division by multiplication.
这一法则使我们能够把 x 单独留在方程的一边。我们总是使用逆运算：加法用减法抵消，减法用加法抵消，乘法用除法抵消，除法用乘法抵消。
Many mistakes happen when students change one side but forget to change the other. Always ask yourself: 'Did I do this to both sides?'
许多错误发生在学生只改变了一边而忘记改变另一边时。始终问自己：“我是否对两边都做了这个运算？”
3. Solving One-Step Equations | 解一步方程
One-step equations require only one inverse operation to find the value of x. There are four basic types: addition, subtraction, multiplication, and division.
一步方程只需要一个逆运算就能求出 x 的值。基本类型有四种：加法、减法、乘法和除法。
Type 1: Addition — Solve x + 5 = 12. Subtract 5 from both sides: x = 12 − 5, so x = 7.
类型 1：加法 — 解 x + 5 = 12。两边同时减去 5：x = 12 − 5，所以 x = 7。
Type 2: Subtraction — Solve x − 4 = 9. Add 4 to both sides: x = 9 + 4, so x = 13.
类型 2：减法 — 解 x − 4 = 9。两边同时加上 4：x = 9 + 4，所以 x = 13。
Type 3: Multiplication — Solve 3x = 18. Divide both sides by 3: x = 18 ÷ 3, so x = 6.
类型 3：乘法 — 解 3x = 18。两边同时除以 3：x = 18 ÷ 3，所以 x = 6。
Type 4: Division — Solve x ÷ 5 = 4. Multiply both sides by 5: x = 4 × 5, so x = 20.
类型 4：除法 — 解 x ÷ 5 = 4。两边同时乘以 5：x = 4 × 5，所以 x = 20。
Once you can recognise the operation attached to x, you simply use the opposite operation on both sides. This is the foundation for all harder equations.
一旦你能识别与 x 相连的运算，只需对两边使用相反的运算即可。这是所有更复杂方程的基础。
4. Solving Two-Step Equations | 解两步方程
A two-step equation has two operations attached to x. You must undo them in the correct order. Usually we undo addition or subtraction first, then multiplication or division.
两步方程有两个与 x 相关的运算。你必须按正确顺序逆运算。通常我们先消除加法或减法，再消除乘法或除法。
3 = 11. Step 1: subtract 3 from both sides: 2x = 8. Step 2: divide both sides by 2: x = 4.
考虑 2x + 3 = 11。第一步：两边同时减去 3：2x = 8。第二步：两边同时除以 2：x = 4。
re is another example: 5x − 7 = 18. Add 7 to both sides: 5x = 25. Then divide by 5: x = 5.
再看一个例子：5x − 7 = 18。两边同时加上 7：5x = 25。然后两边同时除以 5：x = 5。
The order matters because multiplication or division acts on the entire expression, not just on x. If you divide first, you may create fractions that are harder to handle. A good habit is to look for the term being added or subtracted and remove that first.
顺序很重要，因为乘法或除法作用于整个表达式，而不仅仅是 x。如果先做除法，可能会产生更难处理的分式。养成先处理加减项的习惯会更好。
inal equation. For 2x + 3 = 11, substitute x = 4: 2(4) + 3 = 8 + 3 = 11, which is correct.
你可以把答案代回原方程进行检验。对于 2x + 3 = 11，代入 x = 4：2(4) + 3 = 8 + 3 = 11，结果正确。
5. Expanding Brackets in Equations | 方程中去括号
efore solving, you need to expand the brackets or divide both sides by the number outside.
有时线性方程中含有括号，例如 3(x + 2) = 15。求解之前，你需要先去括号，或者在两边同时除以括号外的数。
irst. 3(x + 2) = 15 becomes 3x + 6 = 15. Then subtract 6: 3x = 9. Then divide by 3: x = 3.
方法 1：先展开。3(x + 2) = 15 变为 3x + 6 = 15。然后两边同时减去 6：3x = 9。再两边同时除以 3：x = 3。
e both sides by 3: x + 2 = 5. Then subtract 2: x = 3. Both methods give the same solution.
方法 2：先除以 3。由于两边都能被 3 整除，两边同时除以 3：x + 2 = 5。然后两边同时减去 2：x = 3。两种方法得到相同的解。
x − 4) = 10 expands to 2x − 8 = 10. Add 8 to both sides: 2x = 18. Then divide by 2: x = 9.
当括号内是减法时，要特别注意符号。例如，2(x − 4) = 10 展开为 2x − 8 = 10。两边同时加上 8：2x = 18。然后两边同时除以 2：x = 9。
= 12, expand carefully: −3x − 3 = 12. Add 3 to both sides: −3x = 15. Divide by −3: x = −5.
如果括号前是负数，例如 −3(x + 1) = 12，要小心展开：−3x − 3 = 12。两边同时加上 3：−3x = 15。两边同时除以 −3：x = −5。
6. Equations with Variables on Both Sides | 两边含变量的方程
e first step is to collect all x terms on one side and all number terms on the other side.
有些方程的两边都含有 x，例如 5x + 2 = 3x + 10。第一步是把所有含 x 的项移到一边，把所有数字项移到另一边。
lifies to 2x + 2 = 10. Now subtract 2 from both sides: 2x = 8. Finally divide by 2: x = 4.
首先两边同时减去 3x：5x + 2 − 3x = 3x + 10 − 3x。化简得 2x + 2 = 10。现在两边同时减去 2：2x = 8。最后两边同时除以 2：x = 4。
, subtract 2x from both sides to get 5x − 5 = 10. Then add 5: 5x = 15. Divide by 5: x = 3.
你也可以先移较小的 x 项。对于 7x − 5 = 2x + 10，两边同时减去 2x，得到 5x − 5 = 10。然后两边同时加上 5：5x = 15。两边同时除以 5：x = 3。
Always remember to keep the equation balanced. Moving a term means using the inverse operation on both sides, not just sliding it across and changing the sign without reason.
始终记住要保持方程平衡。移项意味着在两边使用逆运算，而不是简单地把它滑到另一边并随意变号。
e statement such as 3 = 3, every value of x works, and the equation is called an identity.
如果变量项相消后得到一个假命题，如 3 = 5，则方程无解。如果得到一个真命题，如 3 = 3，则任何 x 的值都成立，这样的方程称为恒等式。
7. Equations Involving Fractions | 涉及分数的方程
Many KS3 equations include fractions. The easiest way to solve them is to multiply every term on both sides by the denominator to clear the fraction.
许多 KS3 方程中含有分数。最简单的解法是将两边每一项都乘以分母，以消去分数。
Example 1: x/3 + 2 = 5. Multiply every term by 3: x + 6 = 15. Then subtract 6: x = 9.
例 1：x/3 + 2 = 5。将每一项乘以 3：x + 6 = 15。然后两边同时减去 6：x = 9。
Example 2: 2x/5 = 6. Multiply both sides by 5: 2x = 30. Then divide by 2: x = 15.
例 2：2x/5 = 6。两边同时乘以 5：2x = 30。然后两边同时除以 2：x = 15。
. Multiply every term by 6: 3x + 2x = 30. Combine like terms: 5x = 30. Divide by 5: x = 6.
如果有多个分式，要找出最小公分母。对于 x/2 + x/3 = 5，2 和 3 的最小公分母是 6。将每一项乘以 6：3x + 2x = 30。合并同类项：5x = 30。两边同时除以 5：x = 6。
1 = 3. Multiply by 4: −x + 4 = 12. Subtract 4: −x = 8. Multiply both sides by −1: x = −8.
当分数为负数时，要注意符号。例如，−x/4 + 1 = 3。两边同时乘以 4：−x + 4 = 12。两边同时减去 4：−x = 8。两边同时乘以 −1：x = −8。
8. Checking Your Solution | 检验你的解
Checking is one of the most important habits in algebra. Substitute your value of x into the original equation and confirm that the left side equals the right side.
检验是代数中最重要的习惯之一。把 x 的值代入原方程，确认左边等于右边。
eck by computing 4(4) + 1 = 16 + 1 = 17. Since both sides are 17, the solution is correct.
例如，如果你解 4x + 1 = 17 并得到 x = 4，检验时计算 4(4) + 1 = 16 + 1 = 17。因为两边都是 17，所以解是正确的。
Checking helps you catch mistakes such as sign errors, arithmetic slips, or forgetting to apply an operation to both sides. It also gives you confidence in your final answer.
检验能帮助你发现符号错误、算术失误或忘记对两边进行同一运算等错误。它还能让你对自己的最终答案更有信心。
In an exam, even if you do not have time to check every question, always check equations where you had to deal with negative numbers or fractions, because these are the most likely places for small errors.
在考试中，即使没有时间检查每一道题，也一定要检查涉及负数和分数的方程，因为这些地方最容易出现小错误。
9. Common Mistakes to Avoid | 常见错误及避免方法
Below is a table of frequent errors in solving linear equations and how to correct them.
下表列出了解线性方程时的常见错误及其纠正方法。
| Mistake 错误 | Correction 纠正方法 |
| Only changing one side 只改变一边 | Always do the same operation to both sides 始终对两边做相同运算 |
| Forgetting the sign of a number 忘记数字符号 | Keep the sign with the term when moving it 移项时保留符号 |
| Dividing before adding or subtracting 先除后加减 | Undo addition or subtraction first 先消除加减项 |
| Expanding brackets incorrectly 括号展开错误 | Multiply every term inside the bracket 去括号时乘以括号内每一项 |
| Not checking the answer 不检验答案 | Substitute back into the original equation 代入原方程检验 |
ad of dividing by 3. Remember that 3x means 3 multiplied by x, so the inverse operation is division, not subtraction.
最常见的错误之一是把 3x = 15 写成两边同时减去 3，而不是除以 3。记住 3x 表示 3 乘以 x，所以逆运算是除法，而不是减法。
d 2: x = 10, not x = 6. Pay close attention to whether the number is positive or negative.
另一个错误是移项时丢掉了负号。例如，对于 x + 2 = 8，正确地两边同时减去 2：x = 6。但对于 x − 2 = 8，要两边同时加上 2：x = 10，而不是 x = 6。要密切注意数字是正还是负。
10. Word Problems and Forming Equations | 应用题与列方程
Many exam questions ask you to form an equation from a word problem and then solve it. Start by identifying the unknown and choosing a letter, usually x, to represent it.
许多考试题要求你根据应用题列出方程并求解。首先要找出未知量，并选择一个字母（通常是 x）来表示它。
Example: 'I think of a number, multiply it by 4, then add 7. The result is 31. What is the number?' x. The equation is 4x + 7 = 31. Subtract 7: 4x = 24. Divide by 4: x = 6. The number is 6.
例题：“我想一个数，把它乘以 4，然后加上 7。结果是 31。这个数是多少？”设这个数为 x。方程是 4x + 7 = 31。两边同时减去 7：4x = 24。两边同时除以 4：x = 6。这个数是 6。
Example: ' length of a rectangle is 5 cm more than its width. The perimeter is 38 cm. Find the width.' and: 4x + 10 = 38. Subtract 10: 4x = 28. Divide by 4: x = 7. Width = 7 cm, length = 12 cm.
例题：“一个长方形的长比宽多 5 厘米。周长是 38 厘米。求宽。”设宽为 x。则长为 x + 5。周长：2(x + x + 5) = 38，即 2(2x + 5) = 38。展开：4x + 10 = 38。两边同时减去 10：4x = 28。两边同时除以 4：x = 7。宽为 7 厘米，长为 12 厘米。
test whether you can translate English into algebra. Practice by underlining key phrases: 'more than' means addition, 'less than' means subtraction, 'times' means multiplication, and 'is' or 'equals' means the = sign.
应用题考查你是否能把英文或中文表达转化为代数式。练习时划出关键词：“比……多”表示加法，“比……少”表示减法，“乘以”表示乘法，“是”或“等于”表示等号。
11. Practice Questions with Solutions | 练习题及解答
Try these questions yourself before looking at the solutions. They cover the main types of linear equations at KS3.
在看答案之前，请先自己尝试以下题目。它们涵盖了 KS3 线性方程的主要类型。
- Solve 3x + 4 = 19.
- 解 3x + 4 = 19。
- Solve 5(x − 2) = 20.
- 解 5(x − 2) = 20。
- Solve 7x − 3 = 4x + 9.
- 解 7x − 3 = 4x + 9。
- Solve x/4 + 3 = 7.
- 解 x/4 + 3 = 7。
- Solve 2x/3 = 8.
- 解 2x/3 = 8。
+ 9 → 3x = 12 → x = 4. 4) x/4 + 3 = 7 → x/4 = 4 → x = 16. 5) 2x/3 = 8 → 2x = 24 → x = 12.
解答：1) 3x + 4 = 19 → 3x = 15 → x = 5。2) 5(x − 2) = 20 → x − 2 = 4 → x = 6。3) 7x − 3 = 4x + 9 → 3x = 12 → x = 4。4) x/4 + 3 = 7 → x/4 = 4 → x = 16。5) 2x/3 = 8 → 2x = 24 → x = 12。
When practising, write every step clearly. This makes it easier to find mistakes and helps your teacher or examiner follow your method.
练习时要把每一步写清楚。这样更容易发现错误，也能让老师或阅卷官看懂你的解题方法。
12. Summary and Exam Tips | 总结与考试技巧
Linear equations are a core part of KS3 mathematics. The key skills are using inverse operations, keeping the equation balanced, expanding brackets correctly, handling fractions, and collecting like terms.
线性方程是 KS3 数学的核心内容。关键技能包括使用逆运算、保持方程平衡、正确去括号、处理分数以及合并同类项。
Before an exam, make sure you can solve one-step, two-step, bracket, fraction, and variable-on-both-sides equations confidently. Also practise forming equations from word problems, because these are common in Cambridge Checkpoint tests.
考试前，确保你能熟练地解一步方程、两步方程、含括号方程、含分数方程以及两边含变量的方程。还要练习根据应用题列方程，因为这些在剑桥 Checkpoint 考试中很常见。
During the exam, show all your working. Even if your final answer is wrong, you may receive marks for correct steps. Always check your answer by substituting it back into the original equation.
考试时，要展示所有解题步骤。即使最终答案错了，正确的步骤也可能得分。始终通过代入原方程来检验答案。
Remember that algebra is not about guessing. It is a logical process. If you follow the balancing method step by step, you will be able to solve any linear equation that comes your way.
记住，代数不是靠猜。它是一个有逻辑的过程。只要你一步一步地遵循平衡法，就能解出遇到的任何线性方程。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图