📚 Algebra Essentials: Expressions, Equations and Problem Solving | 代数基础：表达式、方程与问题解决
This Cambridge KS3 maths revision guide focuses on the algebra skills most commonly tested in the lower secondary checkpoint and school assessments. The worked examples build the confidence needed to tackle exercises such as p312_2, where students are asked to simplify, expand, factorise and solve equations.
本剑桥 KS3 数学复习指南聚焦于初中低年级检查和学校评估中最常考的代数技能。所选例题旨在帮助学生建立信心，足以应对如 p312_2 等练习中要求化简、展开、因式分解和求解方程的题目。
1. What is an Algebraic Expression? | 什么是代数表达式？
An algebraic expression uses letters, called variables, to stand for unknown numbers. For example, 3n + 5 means ‘multiply n by 3, then add 5’. Expressions do not contain an equals sign, while equations do.
代数表达式使用称为变量的字母来表示未知数。例如，3n + 5 表示“将 n 乘以 3，再加 5”。表达式不含等号，而方程含有等号。
Key vocabulary: a term is a single number, variable or product such as 4x, -2y or 7; a coefficient is the number part of a term, so in 4x the coefficient is 4; a constant is a term with no variable, such as 5.
关键词汇：项是单个数字、变量或乘积，如 4x、-2y 或 7；系数是项中的数字部分，因此 4x 的系数是 4；常数项是不含变量的项，如 5。
Expression: 3n + 5 | Term: 3n, 5 | Coefficient: 3 | Constant: 5
2. Collecting Like Terms | 合并同类项
Like terms have exactly the same variable part. Only like terms can be added or subtracted. To simplify, group the like terms together and combine their coefficients.
同类项具有完全相同的变量部分。只有同类项才能相加或相减。化简时，将同类项分组并合并其系数。
Example: simplify 4a + 3b – 2a + b. First group the a terms and the b terms, then combine.
示例：化简 4a + 3b – 2a + b。首先将含 a 的项和含 b 的项分组，然后合并。
4a + 3b − 2a + b = (4a − 2a) + (3b + b) = 2a + 4b
Be careful with signs: the minus sign belongs to the term that follows it. In 5x – 2y – x + y, the -x must be grouped with 5x to give 4x, and -2y + y gives -y.
注意符号：减号属于它后面的项。在 5x – 2y – x + y 中，-x 必须与 5x 分组得到 4x，-2y + y 得到 -y。
3. Expanding Brackets | 展开括号
To expand brackets, multiply the term outside the bracket by every term inside. This is the distributive law: a(b + c) = ab + ac.
展开括号时，用括号外的项乘以括号内的每一项。这就是分配律：a(b + c) = ab + ac。
Example: expand 3(2x – 5). Multiply 3 by 2x and then 3 by -5.
示例：展开 3(2x – 5)。将 3 乘以 2x，再乘以 -5。
3(2x − 5) = 3 × 2x + 3 × (−5) = 6x − 15
When the outside multiplier is negative, every sign inside the bracket changes. For example, -2(x + 3) = -2x – 6.
当括号外的乘数为负数时，括号内每一项的符号都要改变。例如，-2(x + 3) = -2x – 6。
4. Factorising Simple Expressions | 因式分解简单表达式
Factorising is the reverse of expanding. You rewrite an expression as a product of a common factor and a bracket. Find the highest common factor of all terms.
因式分解是展开的逆运算。你将一个表达式改写为一个公因数与一个括号的乘积。找出所有项的最大公因数。
Example: factorise 6x + 9. The highest common factor of 6 and 9 is 3, so divide both terms by 3 and put 3 outside the bracket.
示例：因式分解 6x + 9。6 和 9 的最大公因数是 3，因此将两项都除以 3，并把 3 放在括号外。
6x + 9 = 3(2x + 3)
Always check by expanding: 3 × 2x = 6x and 3 × 3 = 9, so the factorised form is correct.
一定要通过展开来检验：3 × 2x = 6x，3 × 3 = 9，因此因式分解的形式正确。
5. Solving One-Step Equations | 解一步方程
To solve an equation, perform the same operation on both sides to isolate the variable. Use inverse operations: addition undoes subtraction, and multiplication undoes division.
解方程时，对等号两边同时进行相同运算以隔离变量。使用逆运算：加法消去减法，乘法消去除法。
Example 1: solve x + 6 = 14. Subtract 6 from both sides.
示例 1：解 x + 6 = 14。两边同时减去 6。
x + 6 − 6 = 14 − 6 → x = 8
Example 2: solve 4x = 20. Divide both sides by 4.
示例 2：解 4x = 20。两边同时除以 4。
4x ÷ 4 = 20 ÷ 4 → x = 5
6. Solving Two-Step Equations | 解两步方程
Two-step equations need two inverse operations. Undo addition or subtraction first, then undo multiplication or division. This reverse order follows the opposite of the order of operations.
两步方程需要两次逆运算。先消去加法或减法，再消去乘法或除法。这个相反顺序与运算顺序的逆序一致。
Example: solve 5x – 3 = 17. Step 1: add 3 to both sides. Step 2: divide both sides by 5.
示例：解 5x – 3 = 17。第一步：两边同时加 3。第二步：两边同时除以 5。
5x − 3 + 3 = 17 + 3 → 5x = 20 → x = 4
For equations of the form x/3 + 2 = 7, subtract 2 first, then multiply by 3: x/3 = 5, so x = 15.
对于 x/3 + 2 = 7 这类方程，先减 2，再乘以 3：x/3 = 5，因此 x = 15。
7. Solving Equations with Brackets | 解含括号的方程
If an equation has brackets, expand them first, then simplify and solve as usual. Expanding first removes the grouping and reveals the like terms.
如果方程含有括号，先展开括号，然后化简并照常求解。先展开可以去除分组并显示出同类项。
Example: solve 2(x + 4) = 18. Expand the left side, then isolate x.
示例：解 2(x + 4) = 18。展开左边，然后隔离 x。
2x + 8 = 18 → 2x = 10 → x = 5
For 3(2x – 1) = 15, expand to 6x – 3 = 15, then add 3 and divide by 6 to get x = 3.
对于 3(2x – 1) = 15，展开得 6x – 3 = 15，然后加 3 并除以 6，得到 x = 3。
8. Solving Equations with Variables on Both Sides | 解未知数在等号两侧的方程
When variables appear on both sides of the equation, collect all variable terms on one side and all constant terms on the other. Use addition or subtraction to move terms across the equals sign.
当未知数出现在等号两侧时，将所有含变量项移到一侧，所有常数项移到另一侧。通过加法或减法将项移过等号。
Example: solve 7x – 2 = 3x + 10. Subtract 3x from both sides, then add 2 to both sides.
示例：解 7x – 2 = 3x + 10。两边同时减去 3x，然后两边同时加 2。
7x − 3x − 2 = 10 → 4x − 2 = 10 → 4x = 12 → x = 3
Always keep the equation balanced: whatever you do to one side, you must do to the other side. This is the golden rule of algebra.
始终保持等式平衡：对一侧进行的任何运算，必须同时对另一侧进行。这是代数学的黄金法则。
9. Word Problems and Forming Equations | 文字题与列方程
To solve word problems, read the question carefully, define the unknown with a letter, and translate the words into an algebraic equation. Then solve using the methods above.
解文字题时，仔细读题，用字母设未知数，并将文字翻译成代数方程。然后使用上述方法求解。
Example: ‘I think of a number, multiply it by 4 and add 7. The answer is 31.’ Let the number be n, so the equation is 4n + 7 = 31.
示例：“我想一个数，将它乘以 4 再加 7，结果是 31。”设这个数为 n，方程就是 4n + 7 = 31。
4n + 7 = 31 → 4n = 24 → n = 6
For a perimeter problem, if a rectangle has length x cm and width 5 cm, and its perimeter is 26 cm, form the equation 2(x + 5) = 26, giving x = 8 cm.
对于周长问题，如果一个矩形的长为 x cm，宽为 5 cm，周长为 26 cm，则可列方程 2(x + 5) = 26，解得 x = 8 cm。
10. Common Mistakes and Checking Answers | 常见错误与检查答案
Common mistakes include forgetting to multiply all terms inside a bracket by a negative sign, only adding a number to one side of the equation, and combining unlike terms such as 3x and 3x².
常见错误包括忘记用负号乘以括号内的所有项，只在一侧加一个数而破坏等式平衡，以及合并非同类的项，如 3x 和 3x²。
Always check your answer by substituting it back into the original equation. For example, if x = 5 solves 3(x – 2) = 9, then 3(5 – 2) = 9 gives 3 × 3 = 9, which is true.
始终将答案代回原方程进行检验。例如，如果 x = 5 是 3(x – 2) = 9 的解，那么 3(5 – 2) = 9 得到 3 × 3 = 9，成立。
Use inverse operations in the reverse order: addition before multiplication is undone by subtraction first and division second. Keep your working neat so you can spot sign errors quickly.
按相反顺序使用逆运算：先加后乘的运算，应先减后除。保持书写工整，以便快速发现符号错误。
11. Practice Strategy for Cambridge KS3 | 剑桥 KS3 练习策略
Build a study routine that mixes fluency practice with problem solving. Start with ten quick simplification exercises, then move to equations of increasing difficulty, and finish with word problems from past papers or textbook exercises such as p312_2.
建立将熟练度练习与问题解决相结合的复习计划。先做十个快速化简练习，然后做难度递增的方程题，最后完成来自历年真题或教科书练习（如 p312_2）的文字题。
Use a checklist: Can I identify like terms? Can I expand brackets with a negative multiplier? Can I factorise by finding the highest common factor? Can I solve equations with variables on both sides? If you answer yes, you are ready for assessment.
使用检查清单：我能识别同类项吗？我能用负乘数展开括号吗？我能通过找最大公因数因式分解吗？我能解未知数在两侧的方程吗？如果回答都是肯定的，你就准备好参加测评了。
- Fluency: 10 simplify-expand-factorise questions daily | 熟练度：每天 10 个化简-展开-因式分解题
- Equations: 8 two-step and bracket equations | 方程：8 个两步方程和含括号方程
- Word problems: 4 real-life context questions weekly | 文字题：每周 4 个实际情境题
12. Summary | 总结
Algebra is a toolkit of symbolic reasoning. Every equation tells a balance story: whatever you do to one side, you do to the other. Mastering expressions and equations is the gateway to higher-level mathematics.
代数是一套符号推理工具。每个方程都在讲述平衡的故事：对一边做什么，就要对另一边做什么。掌握表达式与方程是通往更高层次数学的大门。
Regular practice with careful checking builds accuracy and confidence. Use this guide alongside your Cambridge KS3 materials, and revisit any section where you make mistakes.
定期练习并仔细检查，可以提升准确度和自信心。将本指南与你的剑桥 KS3 教材结合使用，并针对出错的部分进行复习。
Published by TutorHao | Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导