KS3 Maths 代数 变量 表达式 方程 入门
代数（Algebra）是英国 Key Stage 3（7-9年级，11-14岁）数学课程的核心模块，也是 GCSE 数学的基础。它用字母代表未知数，把生活中的数量关系写成简洁的算式。本文用中英双语讲解 KS3 代数基础：变量、表达式、项与系数、化简、代入和解方程，帮助同学们打好 GCSE 的根基。
Algebra is the heart of KS3 maths for students aged 11 to 14 in Year 7 to Year 9. It uses letters to stand for unknown numbers, turning real-life relationships into compact expressions. This bilingual guide covers variables, expressions, terms and coefficients, collecting like terms, substitution, and solving equations, so you can build a solid foundation for GCSE.
1. 什么是代数
代数就是用字母表示数的数学分支。比如「一个苹果加两个苹果等于三个苹果」，如果苹果的价格未知，就可以用字母 p 表示单价。字母 p 在这里叫变量（variable），因为它可以取不同的值。代数让我们不依赖具体数字，也能描述一般规律。
Algebra is the branch of maths in which letters are used to represent numbers. For example, if the price of one apple is unknown, we can use the letter p to stand for it. The letter p is called a variable because its value can change. Algebra lets us describe general rules without relying on specific numbers.
KS3 阶段常见的变量字母有 x, y, n, a, b 等。同一个字母在同一道题里代表同一个数，不同字母可以代表不同数。例如 2x + 3 中，x 是变量，2x 表示「x 的两倍」，3 是常数项（constant term），它的值固定不变。
Common variable letters at KS3 include x, y, n, a and b. In one question, the same letter always represents the same number, while different letters may represent different numbers. In 2x + 3, x is the variable, 2x means "twice x", and 3 is a constant term whose value never changes.
2. 变量与表达式
表达式（expression）是由数字、字母和运算符号组成的组合，例如 3a + 2b - 5。表达式没有等号，所以它不是一个完整的陈述，只是一个「量」。相比之下，含等号的式子叫等式（equation），例如 3a + 2 = 11。
An expression is a combination of numbers, letters and operation signs, such as 3a + 2b - 5. An expression has no equals sign, so it is not a complete statement, just a quantity. A statement with an equals sign, such as 3a + 2 = 11, is called an equation.
写表达式时要熟记简写规则：字母与数字相乘时省略乘号，数字写在字母前面，例如 5 × n 写成 5n；字母相乘省略乘号，例如 a × b 写成 ab；同一字母自乘用指数，例如 x × x 写成 x²。除法通常写成分数形式，例如 x ÷ 3 写成 x/3。
Learn these shorthand rules: when multiplying a number by a letter, drop the multiplication sign and put the number first, so 5 × n becomes 5n; a × b becomes ab; x × x becomes x²; and division is usually written as a fraction, so x ÷ 3 becomes x/3.
3. 项与系数
表达式由若干项（terms）用加减号连接而成。例如 4x² - 3x + 7 有三项：4x²、-3x 和 7。每项前面的数字叫系数（coefficient），4x² 的系数是 4，-3x 的系数是 -3。常数项（如 7）没有字母，系数就是它本身。
An expression is built from terms joined by plus and minus signs. The expression 4x² - 3x + 7 has three terms: 4x², -3x and 7. The number in front of a term is its coefficient: the coefficient of 4x² is 4, and the coefficient of -3x is -3. A constant term such as 7 has no letter, and its coefficient is itself.
同类项（like terms）是指字母部分完全相同的项，例如 5x 与 -2x 是同类项，3xy 与 7xy 是同类项，但 5x 与 5y 不是同类项，因为字母不同。只有同类项才能合并（collect like terms），合并时只加减系数，字母部分保持不变。
Like terms are terms with exactly the same letter part: 5x and -2x are like terms, and 3xy and 7xy are like terms, but 5x and 5y are not, because their letters differ. Only like terms can be collected: you add or subtract their coefficients while the letter part stays unchanged.
4. 化简表达式
化简（simplify）就是合并同类项，把表达式写得最短。例如化简 3x + 5 + 2x - 7：先把含 x 的项放一起，3x + 2x = 5x；再把常数项放一起，5 - 7 = -2；最后结果是 5x - 2。注意运算顺序：加号后的项符号保留，减号后的项符号改变。
To simplify, collect like terms to write the expression as short as possible. For 3x + 5 + 2x - 7: gather the x terms first, 3x + 2x = 5x; then the constants, 5 - 7 = -2; the result is 5x - 2. Keep the sign in front of each term: a minus sign moves with its term.
化简还要会用分配律（distributive law）：a(b + c) = ab + ac。例如化简 3(2x + 4) = 6x + 12，即把括号外的数分别乘进括号内的每一项。若括号外是负数，要特别注意符号，例如 -2(x - 3) = -2x + 6，负负得正。
Simplifying also uses the distributive law: a(b + c) = ab + ac. For example, 3(2x + 4) = 6x + 12: multiply the number outside the bracket by every term inside. Watch the signs when the multiplier is negative: -2(x - 3) = -2x + 6, because a negative times a negative gives a positive.
检查化简结果的小技巧：代入一个数验证。例如 x = 2 时，原式 3x + 5 + 2x - 7 = 6 + 5 + 4 - 7 = 8，化简式 5x - 2 = 10 - 2 = 8，两边相等说明化简正确。这个技巧在考试中非常实用，能避免粗心丢分。
A handy check: substitute a value to verify. With x = 2, the original 3x + 5 + 2x - 7 equals 6 + 5 + 4 - 7 = 8, and the simplified form 5x - 2 equals 10 - 2 = 8. Both sides match, so the simplification is correct. This trick saves many careless marks in exams.
5. 解方程入门
方程是含等号的代数陈述，例如 2x + 5 = 13。解方程（solve the equation）就是找出使等式成立的未知数的值。核心方法是「逆运算」（inverse operations）：加用减消去，乘用除去消去，两边始终做同样的运算，保持天平平衡。
An equation is an algebraic statement with an equals sign, such as 2x + 5 = 13. Solving an equation means finding the value of the unknown that makes it true. The key idea is inverse operations: undo addition with subtraction and multiplication with division, always applying the same operation to both sides to keep the balance.
一步方程直接逆运算：x + 7 = 15，两边同时减 7，得 x = 8；3x = 21，两边同时除以 3，得 x = 7。两步方程要按顺序处理：2x + 5 = 13，先两边减 5 得 2x = 8，再两边除以 2，得 x = 4。最后把 x = 4 代回原式检查：2 × 4 + 5 = 13，正确。
One-step equations use a single inverse: x + 7 = 15, subtract 7 from both sides to get x = 8; 3x = 21, divide both sides by 3 to get x = 7. Two-step equations are done in order: for 2x + 5 = 13, first subtract 5 to get 2x = 8, then divide by 2 to get x = 4. Check by substitution: 2 × 4 + 5 = 13, correct.
还有两类常见方程：含括号的方程，先展开再解，例如 3(x - 1) = 12，展开得 3x - 3 = 12，解得 x = 5；未知数在两侧的方程，先把含未知数的项移到同一边，例如 4x - 3 = 2x + 9，两边减 2x 得 2x - 3 = 9，再解得 x = 6。
Two more common types: equations with brackets, expand first then solve, such as 3(x - 1) = 12, which becomes 3x - 3 = 12, giving x = 5; and equations with unknowns on both sides, move all unknown terms to one side first, such as 4x - 3 = 2x + 9, subtract 2x from both sides to get 2x - 3 = 9, then x = 6.
6. 代入与函数入门
代入（substitution）是把已知数值放进表达式求值。例如表达式 3a - b，当 a = 5、b = 2 时，代入得 3 × 5 - 2 = 13。代入时注意：数字与字母相乘要写乘号，负数要加括号，例如 a = -3 时代入 2a 得 2 × (-3) = -6。
Substitution means putting given values into an expression. For 3a - b with a = 5 and b = 2, substitute to get 3 × 5 - 2 = 13. Write the multiplication sign explicitly and put brackets around negatives: with a = -3, 2a becomes 2 × (-3) = -6.
函数（function）是 KS3 后期的重点：它像一个机器，输入一个数，按规则输出另一个数。函数可以用箭头图表示：x -> 2x + 1，表示输入 x，输出 2x + 1。当 x = 3 时输出 7，当 x = 10 时输出 21。理解函数有助于后续学习直线图像 y = mx + c。
Functions are a key topic in later KS3: a function acts like a machine that takes an input and produces an output by a rule. A function machine can be shown as x -> 2x + 1: input x, output 2x + 1. When x = 3 the output is 7, and when x = 10 the output is 21. Understanding functions prepares you for straight-line graphs such as y = mx + c.
7. 用代数解决实际问题
KS3 考试中代数经常以应用题（word problems）的形式出现。解题四步法：第一步读题，找出未知量并用字母表示，例如「令 Anna 有 x 颗糖」；第二步根据题意列方程；第三步解方程求 x；第四步把答案代回题目检查是否符合常理。这四步在答题纸上写清楚，能拿满分方法分。
In KS3 exams algebra often appears as word problems. Follow four steps: first read the question and define the unknown, for example "let Anna have x sweets"; second build an equation from the information given; third solve the equation; fourth check the answer makes sense in the original story. Writing these steps clearly earns full method marks.
看一个例子：Anna 和 Ben 一共有 30 颗糖，Ben 比 Anna 多 4 颗，Anna 有多少颗？设 Anna 有 x 颗，则 Ben 有 x + 4 颗，列方程 x + (x + 4) = 30，化简得 2x + 4 = 30，两边减 4 得 2x = 26，除以 2 得 x = 13。检查：13 + 17 = 30，符合题意，Anna 有 13 颗糖。
Worked example: Anna and Ben have 30 sweets in total, and Ben has 4 more than Anna. How many does Anna have? Let Anna have x, so Ben has x + 4. The equation is x + (x + 4) = 30, which simplifies to 2x + 4 = 30. Subtract 4 from both sides: 2x = 26. Divide by 2: x = 13. Check: 13 + 17 = 30, which fits the story, so Anna has 13 sweets.
常见的设未知数陷阱：题目问「每个班有多少学生」时，直接设 x 为学生数，不要设成班级数；题目出现「比…多 3 倍」时，注意是 3x 还是 x + 3，多 3 倍是 3x，多 3 个才是 x + 3。把中文题意翻译成代数式是 KS3 的核心能力，值得专门练习。
Common traps when defining unknowns: if the question asks for the number of students in each class, let x be the number of students, not the number of classes; "three times as many" means 3x, while "three more" means x + 3. Translating English or Chinese wording into algebra is a core KS3 skill, so practise it deliberately.
8. 不等式入门
不等式（inequality）用符号比较两个量的大小：大于 >，小于 <，大于等于 ≥，小于等于 ≤。例如 x > 3 表示 x 是大于 3 的任意数，它和方程 x = 3 完全不同：方程只有一个解，而不等式有无数个解。表示解集时常用数轴（number line），开圆点表示不含端点，实心圆点表示包含端点。
An inequality compares two quantities with symbols: greater than >, less than <, greater than or equal to ≥, and less than or equal to ≤. For example x > 3 means any number larger than 3, which is very different from the equation x = 3: an equation has one solution, while an inequality has infinitely many. Solutions are often shown on a number line, with an open circle for an excluded endpoint and a filled circle for an included one.
解不等式的方法和解方程类似：两边可以同时加减同一个数，也可以同时乘除同一个正数。唯一要注意的是：两边同时乘除负数时，不等号方向必须翻转。例如解 -2x > 6，两边除以 -2，不等号翻转，得 x < -3。验证：取 x = -4，-2 × (-4) = 8 > 6，成立。
Solving inequalities is like solving equations: you can add or subtract the same number on both sides, and multiply or divide by the same positive number. The one special rule: when you multiply or divide both sides by a negative number, you must flip the inequality sign. For example, to solve -2x > 6, divide both sides by -2 and flip the sign to get x < -3. Check with x = -4: -2 × (-4) = 8 > 6, which holds.
再解一个两步不等式：2x + 1 ≤ 9。先两边减 1 得 2x ≤ 8，再两边除以 2 得 x ≤ 4。意思是 x 可以是 4 或任何小于 4 的数，例如 3.5、0、-100 都满足。不等式在 GCSE 中会与数轴、区间和实际情境（如「至少」「最多」）结合出题，KS3 打好符号基础非常重要。
One more two-step example: 2x + 1 ≤ 9. Subtract 1 from both sides: 2x ≤ 8. Divide by 2: x ≤ 4. This means x can be 4 or any number less than 4, such as 3.5, 0 or -100. At GCSE, inequalities combine with number lines, intervals and real-life phrases like "at least" and "at most", so mastering the symbols at KS3 really matters.
9. 常见错误与考试技巧
KS3 代数最常见的错误有四个：第一，把 2x 当成 2 + x，其实 2x 是 2 × x；第二，合并非同类项，例如把 2x + 3 合并成 5x，这是错的，2x 与 3 不能合并；第三，漏掉负号，例如 5 - 2x 减 3x 时把结果写成 5 + x；第四，忘记两边同操作，只在一边做运算破坏方程平衡。
The four most common KS3 algebra mistakes: first, treating 2x as 2 + x, when it actually means 2 × x; second, collecting unlike terms, such as wrongly combining 2x + 3 into 5x; third, losing negative signs, for example writing 5 + x when subtracting 3x from 5 - 2x; fourth, operating on only one side of an equation, breaking its balance.
考试技巧：每题先写「令 x 代表…」，展示思考过程；解方程最后一步一定代回验证；化简题用代入法检查；含分数的题目先通分再合并。这些习惯能让阅卷老师看清步骤，也能帮你拿满方法分（method marks），即使最后答案算错也能保住大部分分数。
Exam tips: state what x stands for at the start to show your thinking; always check your solution by substituting it back; verify simplifications by plugging in a number; for fractions, find a common denominator first. Clear working earns method marks, so even a wrong final answer keeps most of the credit.
KS3 代数基础虽然概念简单，但它是 GCSE 数学、物理和经济等科目的语言。每天做 10 分钟代数练习，坚持一学期，你的计算速度和准确率会有明显提升。祝学习顺利！
KS3 algebra may look simple, but it is the language of GCSE maths, physics and economics. Ten minutes of algebra practice every day for one term will noticeably improve both your speed and accuracy. Good luck with your studies!
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