📚 Algebraic Expressions and Equations | 代数表达式与方程
Algebra is the branch of mathematics where we use letters and symbols to represent numbers and quantities in formulas and equations. In this article, you will learn how to write, simplify, and solve algebraic expressions and equations, starting from the very basics up to multi‑step problems suitable for Key Stage 3 learners following the Cambridge curriculum.
代数是数学的一个分支，我们用字母和符号来表示公式和方程中的数字与数量。本文将从最基础的知识讲起，教大家如何书写、化简和求解代数表达式与方程，内容循序渐进，直至包含适合剑桥课程 KS3 阶段的多步问题。
1. Understanding Variables and Constants | 理解变量与常数
A variable is a symbol, usually a letter like x or y, that stands for an unknown number or a value that can change. A constant is a fixed number, such as 3, −5, or ½, that never changes its value.
变量是一个符号，通常用 x 或 y 这样的字母表示，它代表一个未知数或可以变化的值。常数是一个固定的数，例如 3、−5 或 ½，其值始终不变。
In the expression 4n + 7, n is the variable and 4 and 7 are constants. The number 4 is also called a coefficient because it multiplies the variable n.
在表达式 4n + 7 中，n 是变量，4 和 7 是常数。数字 4 也被称为系数，因为它与变量 n 相乘。
Every term in an algebraic expression has three parts: a sign (+ or −), a number (the coefficient), and a variable part (which may have an exponent).
代数表达式中的每一项都有三个部分：符号（+ 或 −）、数字（系数）以及变量部分（可能带有指数）。
2. Writing Algebraic Expressions from Words | 将文字表述转化为代数表达式
To translate a word problem into algebra, link key words to operations: ‘sum’ means addition, ‘difference’ means subtraction, ‘product’ means multiplication, and ‘quotient’ means division. The phrase ‘a number’ is usually represented by a variable such as n.
要将文字题转化为代数，需要把关键词与运算联系起来：“和”表示加法，“差”表示减法，“积”表示乘法，“商”表示除法。短语“某个数”通常用 n 这样的变量表示。
- English: “Three more than a number” → n + 3
- 中文： “比某个数大 3” → n + 3
- English: “Twice a number decreased by five” → 2n − 5
- 中文： “一个数的两倍减去 5” → 2n − 5
- English: “The quotient of a number and 4” → n ÷ 4, written as n/4
- 中文： “某个数除以 4” → n ÷ 4，写作 n/4
Practice converting both simple and compound phrases so you can build expressions fluently.
请练习转换简单短语和复合短语，以便能流畅地构建表达式。
3. Like Terms and Simplifying Expressions | 同类项与表达式的化简
Like terms have exactly the same variable part raised to the same power; for example, 3x and −2x are like terms, whereas 3x and 3x² are not. To simplify an expression, add or subtract the coefficients of like terms while keeping the variable part unchanged.
同类项是指变量部分完全相同且指数也相同的项；例如，3x 和 −2x 是同类项，而 3x 与 3x² 则不是。化简表达式时，只需将同类项的系数相加或相减，变量部分保持不变。
Example: 5a + 2b − 3a + 4b = (5 − 3)a + (2 + 4)b = 2a + 6b
示例： 5a + 2b − 3a + 4b = (5 − 3)a + (2 + 4)b = 2a + 6b
Always reorder terms so the answer is written in alphabetical order of the variables, with any constant term last.
总是重新排列各项，使答案按变量的字母顺序书写，常数项放在最后。
4. Expanding Brackets Using the Distributive Law | 利用分配律展开括号
ltiply the term outside the bracket by every term inside, paying close attention to the signs.
分配律指出，a(b + c) = a × b + a × c。减法时同样适用：a(b − c) = a × b − a × c。用括号外面的项乘以括号内的每一项，并特别注意符号。
Example: Expand 3(2x − 5) → 3 × 2x + 3 × (−5) = 6x − 15
示例： 展开 3(2x − 5) → 3 × 2x + 3 × (−5) = 6x − 15
negative sign in front of the bracket, treat it as multiplying by −1: −(4y + 3) = −4y − 3.
如果括号前有负号，可将其视为乘以 −1：−(4y + 3) = −4y − 3。
5. Factorising Simple Expressions | 简单表达式的因式分解
Factorising is the reverse of expanding. You look for the highest common factor (HCF) of the terms and write it outside a bracket, with the remaining factors inside.
因式分解是展开的逆过程。你需要找出各项的最大公因数（HCF），将其写在括号外面，剩下的因子放在括号内。
Example: Factorise 12x + 8 → HCF of 12 and 8 is 4, so 4(3x + 2).
示例： 对 12x + 8 进行因式分解 → 12 和 8 的最大公因数是 4，因此 4(3x + 2)。
Always multiply back mentally to check that your bracket gives the original expression.
一定要在心里回乘一遍，检验括号内表达式是否与原式一致。
6. Solving One‑Step Equations | 解一步方程
An equation states that two expressions are equal. To solve an equation, you must keep the balance by performing the same operation on both sides. The goal is to isolate the variable.
方程表示两个表达式相等。解方程时，必须通过对方程两边执行相同的运算来保持平衡，目标是求出变量。
- Addition equation: x − 7 = 12 → add 7 to both sides → x = 19
- 中文： 加法方程：x − 7 = 12 → 两边加 7 → x = 19
- Multiplication equation: 5y = 45 → divide both sides by 5 → y = 9
- 中文： 乘法方程：5y = 45 → 两边除以 5 → y = 9
Always write your final answer as x = … and, where time allows, substitute back to check.
最终答案务必写成 x = … 的形式，并且时间允许的话，将答案代入原方程检验。
7. Solving Two‑Step Equations | 解两步方程
A two‑step equation involves two operations. Undo them in reverse order: first deal with addition or subtraction, then multiplication or division.
两步方程包含两种运算。请按照相反的运算顺序求解：先处理加减，再处理乘除。
Example: 3x + 4 = 19
示例： 3x + 4 = 19
3x + 4 = 19
3x = 19 − 4
3x = 15
x = 5
Make sure to show each step clearly so an examiner can follow your reasoning.
请确保清晰地写出每一步，以便阅卷人跟随你的推理过程。
8. Equations with Brackets | 带括号的方程
When an equation contains brackets, always expand them first before applying inverse operations. This transforms the equation into a simpler, two‑step or multi‑step form.
当方程中含有括号时，一定要先展开括号，再应用逆运算。这样就能将方程转化成更简单的两步或多步形式。
Example: 2(3x − 1) = 10
示例： 2(3x − 1) = 10
6x − 2 = 10
6x = 12
x = 2
After expanding, proceed exactly like a standard two‑step equation.
展开后，就像标准的两步方程一样继续求解。
9. Equations with Variables on Both Sides | 变量在等号两边的方程
If the variable appears on both sides of the equals sign, start by eliminating it from one side. Choose the side with the smaller coefficient to move, using addition or subtraction.
如果变量同时出现在等号两边，首先要把它从其中一边消去。选择系数较小的一边，通过加法或减法将其移项。
Example: 5x + 2 = 3x + 10
示例： 5x + 2 = 3x + 10
5x − 3x + 2 = 10
2x = 8
x = 4
This method keeps the coefficient of x positive and reduces the number of steps.
这种方法可以让 x 的系数保持为正，并减少步骤。
10. Forming Equations from Word Problems | 根据文字题建立方程
To turn a real‑life problem into an equation, give the unknown quantity a variable name, then translate each piece of information into algebraic language. Finally, write the relationship as an equation and solve.
要将实际问题转化为方程，首先给未知量设定一个变量名称，然后将每一个信息转化为代数语言。最后把等量关系写成方程并求解。
Word problem: The sum of three consecutive numbers is 72. Find the numbers.
文字题： 三个连续整数的和是 72，求这三个数。
Let the first number be n. The next two are n + 1 and n + 2.
设第一个数为 n，那么后两个数分别是 n + 1 和 n + 2。
n + (n + 1) + (n + 2) = 72
3n + 3 = 72
3n = 69
n = 23
The three numbers are 23, 24, and 25.
这三个数是 23、24 和 25。
11. Using Formulae and Substitution | 公式的运用与代入法
e a formula, replace the given variables with their numbers and carry out the calculation.
公式是一种特殊的方程，用来表示两个或多个变量之间的关系，例如长方形的面积公式 A = l × w。要计算公式的值，只需将给定的变量替换为其数值并进行运算。
Example: If v = u + at, find v when u = 2, a = 3.5, and t = 4.
示例： v = u + at，已知 u = 2，a = 3.5，t = 4，求 v。
v = 2 + (3.5 × 4) = 2 + 14 = 16
Always follow BIDMAS (brackets, indices, division/multiplication, addition/subtraction) when substituting.
代入时务必遵循运算顺序（先括号，再指数，然后乘除，最后加减）。
12. Common Mistakes and How to Avoid Them | 常见错误及避免方法
| Mistake / 错误 | Correction / 纠正 |
|---|---|
| Not distributing a negative sign: −(x − 3) = −x − 3 未将负号分配：−(x − 3) = −x − 3 | −(x − 3) = −x + 3 |
| Adding unlike terms: 2a + 3b = 5ab 将不同类项相加：2a + 3b = 5ab | 2a + 3b stays as 2a + 3b 2a + 3b 仍需保持为 2a + 3b |
| Ignoring the balance method: doing something to only one side 忽视平衡法：只对等式一边进行运算 | Always apply the operation to both sides. 始终对等号两边同时运算。 |
Build the habit of checking each line as you write it, and read the question again to ensure your answer makes sense in the original context.
养成边写边检查的习惯，并重新读题，确保你的答案在原始情境中是合理的。
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