📚 Cambridge KS3 Mathematics: Solving Linear Equations | 剑桥KS3数学：解线性方程
Linear equations are the foundation of algebra at KS3 level. In Cambridge Lower Secondary Mathematics, students must learn how to solve equations using inverse operations, balance both sides of an equation, and apply these skills to real-world problems. This revision article explains the key methods step by step, with English and Chinese explanations paired together so that learners can build confidence and accuracy.
线性方程是 KS3 阶段代数的基础。在剑桥初中数学中，学生必须学会使用逆运算解方程、保持等式两边平衡，并将这些技能应用于实际问题。本篇复习文章逐步讲解关键方法，英文和中文解释成对出现，帮助学习者建立信心并提高准确性。
1. What Is a Linear Equation? | 什么是线性方程？
uations can appear in many forms, but they can all be solved by undoing operations in the correct order.
线性方程是指未知数（通常写作 x）的次数为 1 的方程。线性方程的图像总是一条直线。例如 x + 5 = 11、3x − 2 = 10 和 2(x + 4) = 18。线性方程可以有多种形式，但都可以按照正确顺序逆推运算来求解。
The word 'linear' comes from 'line', because every solution set forms a straight line when plotted on a coordinate grid. At KS3, you do not need to graph every equation, but understanding this link helps you see why the methods work.
"线性" 一词来自 "直线"，因为每个解集在坐标网格上绘制时都形成一条直线。在 KS3 阶段，你不需要画出每个方程的图像，但理解这种联系有助于你明白这些方法为何有效。
2. The Balancing Method | 天平平衡法
Think of an equation as a balance scale. The left side and the right side must always have the same value. If you add, subtract, multiply, or divide one side by a number, you must do exactly the same to the other side. This is called the balancing method, and it is the most reliable way to solve linear equations.
把方程想象成一个天平。左边和右边必须始终具有相同的值。如果你对一边加上、减去、乘以或除以某个数，就必须对另一边做完全相同的操作。这被称为天平平衡法，是解线性方程最可靠的方法。
the left side is x + 3 and the right side is 10. To isolate x, subtract 3 from both sides:
例如，在方程 x + 3 = 10 中，左边是 x + 3，右边是 10。为了分离出 x，两边同时减去 3：
x + 3 − 3 = 10 − 3 → x = 7
your answer back into the original equation. Here, 7 + 3 = 10, so the solution is correct.
始终将答案代回原方程进行检验。这里 7 + 3 = 10，所以解是正确的。
3. Solving One-Step Equations | 解一步方程
One-step equations require only one inverse operation to find the unknown. The four possible operations are addition, subtraction, multiplication, and division. The key is to perform the opposite operation on both sides.
一步方程只需要一次逆运算就能求出未知数。四种可能的运算是加法、减法、乘法和除法。关键是要在两边同时执行相反的运算。
- Addition: x + 4 = 9 → subtract 4 → x = 5. 加法：x + 4 = 9 → 减去 4 → x = 5。
- Subtraction: x − 6 = 3 → add 6 → x = 9. 减法：x − 6 = 3 → 加上 6 → x = 9。
- Multiplication: 5x = 35 → divide by 5 → x = 7. 乘法：5x = 35 → 除以 5 → x = 7。
- Division: x ÷ 2 = 8 → multiply by 2 → x = 16. 除法：x ÷ 2 = 8 → 乘以 2 → x = 16。
In each case, the inverse operation undoes the original operation. Multiplication and division are inverse operations, just as addition and subtraction are inverse operations.
在每种情况下，逆运算都会撤销原来的运算。乘法和除法互为逆运算，加法和减法也互为逆运算。
4. Solving Two-Step Equations | 解两步方程
A two-step equation has two operations attached to the unknown. You must undo them in reverse order, like taking off a coat before removing a jumper. Normally, you undo addition or subtraction first, then multiplication or division.
两步方程的未知数包含两步运算。你必须按相反顺序逐步撤销，就像先脱掉外套再脱掉毛衣一样。通常先处理加法或减法，再处理乘法或除法。
The x has been multiplied by 3 and then 2 has been subtracted. First add 2 to both sides:
例如，解 3x − 2 = 13。x 先被乘以 3，然后减去了 2。首先两边同时加上 2：
3x − 2 + 2 = 13 + 2 → 3x = 15
Then divide both sides by 3:
然后两边同时除以 3：
3x ÷ 3 = 15 ÷ 3 → x = 5
15 − 2 = 13, which is correct. This reverse-order method works for all two-step equations.
检验：3(5) − 2 = 15 − 2 = 13，正确。这种逆序方法适用于所有两步方程。
5. Equations with Brackets | 含括号的方程
he number or term outside. For example, 2(x + 4) means 2 × x + 2 × 4, which equals 2x + 8.
当线性方程含有括号时，通常应先去括号。去括号是指将括号内的每一项乘以括号外的数或项。例如，2(x + 4) 表示 2 × x + 2 × 4，等于 2x + 8。
Solve 2(x + 4) = 18. First expand the left side:
解方程 2(x + 4) = 18。首先展开左边：
2x + 8 = 18
Then subtract 8 from both sides and divide by 2:
然后两边同时减去 8，再除以 2：
2x = 10 → x = 5
4 = 9, then subtract 4. Both approaches give x = 5. Choose the method that feels clearest.
或者，你也可以先两边同时除以 2，得到 x + 4 = 9，再减去 4。两种方法都能得到 x = 5。选择你觉得最清晰的方法。
6. Unknowns on Both Sides | 未知数在等号两边
first goal is to collect the x terms on one side and the number terms on the other side. You can do this by adding or subtracting the same term from both sides.
一些 KS3 方程的未知数出现在等号两边，例如 5x + 3 = 3x + 11。首要目标是把含 x 的项移到一边，把数字项移到另一边。你可以通过在两边同时加上或减去相同的项来实现。
Start by subtracting 3x from both sides:
首先从两边同时减去 3x：
5x + 3 − 3x = 3x + 11 − 3x → 2x + 3 = 11
Now the equation is a familiar two-step equation:
现在这个方程就变成了熟悉的两步方程：
2x + 3 = 11 → 2x = 8 → x = 4
side 5(4) + 3 = 23, right side 3(4) + 11 = 23. Both sides are equal, so x = 4 is correct.
检验：左边 5(4) + 3 = 23，右边 3(4) + 11 = 23。两边相等，所以 x = 4 正确。
7. Equations with Fractions | 含分数的方程
clears the fractions and leaves a simpler linear equation. For example, solve x/3 + 1 = 5.
含有分数的方程可能看起来有难度，但如果你把每一项都乘以最小公分母，就会变得简单。这样可以消去分母，留下更简单的线性方程。例如，解 x/3 + 1 = 5。
Multiply every term by 3:
将每一项都乘以 3：
3 × (x/3) + 3 × 1 = 3 × 5 → x + 3 = 15
n subtract 3 from both sides to get x = 12. Check: 12/3 + 1 = 4 + 1 = 5, which is correct.
然后两边同时减去 3，得到 x = 12。检验：12/3 + 1 = 4 + 1 = 5，正确。
every term by 6, the lowest common multiple of 2 and 3. This gives 3x = 2x + 6, so x = 6.
当有两个分数时，例如 x/2 = x/3 + 1，将每一项乘以 6（2 和 3 的最小公倍数）。得到 3x = 2x + 6，所以 x = 6。
8. Forming Equations from Word Problems | 从应用题建立方程
KS3 exams often ask you to form an equation from a written problem. Read the problem carefully, identify the unknown, and give it a letter. Then translate the words into algebraic expressions and set up an equation.
KS3 考试经常要求你根据文字题建立方程。仔细阅读题目，找出未知数，并用字母表示。然后把文字翻译成代数式，并建立方程。
the width be x cm. Then the length is 2x + 3 cm. The perimeter is 2(length + width) = 42:
示例：一个长方形的长比宽的 2 倍多 3 厘米。周长是 42 厘米。求宽。设宽为 x 厘米。则长为 2x + 3 厘米。周长为 2(长 + 宽) = 42：
2((2x + 3) + x) = 42 → 2(3x + 3) = 42 → 6x + 6 = 42
Subtract 6 and divide by 6:
减去 6 再除以 6：
6x = 36 → x = 6
Always re-read the question to make sure your answer makes sense in the original context.
宽为 6 厘米，长为 2(6) + 3 = 15 厘米。始终重新读题，确保答案在原始情境中合理。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Many students lose marks on linear equations because of small slips. The most common mistakes include only operating on one side, mixing up inverse operations, forgetting to change signs when moving terms, and expanding brackets incorrectly.
许多学生在解线性方程时因为小失误而丢分。最常见的错误包括只对一边进行运算、混淆逆运算、移项时忘记变号，以及去括号时展开错误。
- Only one side changed: x + 2 = 6 → x = 6 + 2 is wrong. You must subtract 2 from both sides. 只改变一边：x + 2 = 6 → x = 6 + 2 是错误的。你必须从两边同时减去 2。
- Wrong inverse: 3x = 12 → x = 12 × 3 is wrong. You should divide by 3. 逆运算错误：3x = 12 → x = 12 × 3 是错误的。你应该除以 3。
- Sign errors: When subtracting 2x from 5x, the result is 3x, not 7x. 符号错误：从 5x 中减去 2x，结果是 3x，而不是 7x。
- Bracket expansion: 3(x + 4) = 3x + 4 is wrong. It should be 3x + 12. 括号展开：3(x + 4) = 3x + 4 是错误的。应为 3x + 12。
Always write each step clearly, and check your final answer by substitution. If your check does not work, go back through your working line by line.
始终清晰地写出每一步，并通过代入检验最终答案。如果检验不成立，就逐行回顾你的解题过程。
10. Practice Questions with Worked Solutions | 练习题与解答示范
Try these practice questions before looking at the solutions. Cover the answers, solve each equation, then check your work step by step.
在查看答案之前先尝试这些练习题。遮住答案，解出每个方程，然后逐步核对你的解题过程。
Question 1: Solve 4x + 7 = 31. 题目 1：解 4x + 7 = 31。
Subtract 7 from both sides: 4x = 24. Divide by 4: x = 6. Check: 4(6) + 7 = 24 + 7 = 31. 两边同时减去 7：4x = 24。除以 4：x = 6。检验：4(6) + 7 = 24 + 7 = 31。
Question 2: Solve 5(x − 2) = 3x + 4. 题目 2：解 5(x − 2) = 3x + 4。
2x − 10 = 4. Add 10: 2x = 14. Divide by 2: x = 7. Check: 5(7 − 2) = 25 and 3(7) + 4 = 25. 展开左边：5x − 10 = 3x + 4。减去 3x：2x − 10 = 4。加上 10：2x = 14。除以 2：x = 7。检验：5(7 − 2) = 25，3(7) + 4 = 25。
Question 3: Solve x/4 + 3 = 8. 题目 3：解 x/4 + 3 = 8。
Subtract 3: x/4 = 5. Multiply by 4: x = 20. Check: 20/4 + 3 = 5 + 3 = 8. 减去 3：x/4 = 5。乘以 4：x = 20。检验：20/4 + 3 = 5 + 3 = 8。
11. Using a Table to Organise Steps | 使用表格整理步骤
A table can help you track the balancing method clearly, especially when an equation has several operations. Write the equation, the operation you will perform, and the new equation after each step.
表格可以帮助你清晰地记录天平平衡法的步骤，尤其是当方程包含多个运算时。写出方程、将要执行的运算以及每一步之后的新方程。
| Step 步骤 | Equation 方程 | Operation 运算 |
|---|---|---|
| Start 开始 | 6x − 4 = 2x + 12 | — |
| 1 | 6x − 4 − 2x = 2x + 12 − 2x | Subtract 2x 减去 2x |
| 2 | 4x − 4 = 12 | Simplify 化简 |
| 3 | 4x − 4 + 4 = 12 + 4 | Add 4 加上 4 |
| 4 | 4x = 16 | Simplify 化简 |
| 5 | x = 4 | Divide by 4 除以 4 |
Using a table like this reduces the risk of missing a step or changing only one side of the equation. It also makes revision easier because you can see the logical flow at a glance.
使用这样的表格可以降低遗漏步骤或只改变方程一边的风险。它还能让复习更轻松，因为你可以一眼看清逻辑流程。
12. Summary and Revision Tips | 总结与复习建议
To master linear equations at KS3, remember these key points: use inverse operations, keep both sides balanced, undo operations in reverse order, expand brackets before collecting like terms, and always substitute your answer to check it. Practice a little every day rather than cramming all at once.
要在 KS3 阶段掌握线性方程，请记住这些要点：使用逆运算、保持两边平衡、按相反顺序撤销运算、先展开括号再合并同类项，并且始终代入答案进行检验。每天练习一点，而不是临时死记硬背。
When revising, create a checklist of equation types: one-step, two-step, brackets, unknowns on both sides, and fractions. Try to solve at least one of each type without notes. Mark your work, correct any mistakes, and redo weak areas until they become automatic.
复习时，制作一份方程类型清单：一步方程、两步方程、含括号方程、未知数在两边和含分数方程。尝试在不看笔记的情况下每种类型至少解一道题。批改作业，纠正错误，并重做薄弱环节，直到它们变得熟练自如。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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