📚 Mastering Algebra: Expressions and Equations | 代数初步：表达式与方程
Algebra is like a new language that helps us describe patterns, solve puzzles and understand relationships between numbers. In KS3 Cambridge Mathematics, you will learn to write expressions, simplify them and solve equations using a step-by-step approach. This article will guide you through the key ideas, from collecting like terms to tackling word problems, giving you plenty of practice and clear explanations.
代数就像一门新语言，它帮助我们描述规律、解开谜题并理解数字之间的关系。在剑桥初中数学中，你将学习如何写出代数表达式、化简式子以及用逐步计算的方法解方程。本文将从合并同类项到处理应用题，一一讲解核心概念，并提供丰富的练习和清晰的解释。
1. What is algebra? | 什么是代数？
Algebra uses letters, often called variables, to stand for unknown numbers. For example, if I think of a number and add 5 to it, I can write the expression n + 5. The letter n can take different values, which makes algebra flexible and powerful. In KS3, you move from using blanks in primary school maths to working confidently with symbols like x, y and a.
代数使用字母（通常称为变量）来表示未知数。例如，如果我想一个数再加上5，就可以写成表达式 n + 5。字母 n 可以代表不同的数值，这让代数既灵活又强大。在初中阶段，你会从小学时使用的空格过渡，开始熟练使用 x、y 和 a 等符号。
2. Variables, constants and terms | 变量、常数和项
In an algebraic expression, each part separated by a + or – sign is called a term. A term can be just a number, like 7 (a constant), a letter alone, like x, or a number and letter multiplied together, like 3y. The number part is called the coefficient. Understanding these building blocks helps you manipulate expressions correctly.
在代数表达式中，由加号或减号分隔的每一部分都叫一个项。项可以只是一个数字，如 7（常数），可以只是一个字母，如 x，也可以是数字和字母相乘，如 3y。数字部分叫做系数。理解这些基本构件有助于你正确地操作表达式。
3. Collecting like terms | 合并同类项
Like terms have exactly the same variable letters raised to the same power. You can add or subtract them just as you combine apples with apples. For instance, 5x and 3x are like terms, giving 8x when added. However, 2x² and 3x are not like terms because the powers differ. This skill makes long expressions much neater.
同类项是指那些含有完全相同的字母且相同次数的项。你可以像把苹果和苹果放在一起一样来加减它们。例如，5x 和 3x 是同类项，相加得 8x。但是 2x² 和 3x 不是同类项，因为次数不同。掌握这个技巧能让冗长的表达式变得整洁许多。
Simplify by collecting like terms: 4a + 3b – a + 7b → (4a – a) + (3b + 7b) = 3a + 10b. Always circle or underline like terms first to avoid mistakes.
通过合并同类项化简：4a + 3b – a + 7b → (4a – a) + (3b + 7b) = 3a + 10b。始终先圈出或下划线标出同类项，以免出错。
4. Expanding brackets | 去括号展开
When you see a number or term directly outside a bracket, you must multiply everything inside by it. This process is called expanding. For example, 3(2x + 5) expands to 6x + 15. Always use the distributive law: multiply each term in the bracket by the term outside. Double brackets, like (x + 2)(x + 3), need careful expansion using the FOIL method.
当你看到括号外直接有一个数字或项时，必须用它乘以括号内的每一项，这个过程叫作展开。例如，3(2x + 5) 展开后为 6x + 15。务必使用分配律：用外边的项去乘括号里的每一项。像 (x + 2)(x + 3) 这样的双重括号则需要使用 FOIL 方法仔细展开。
Expand: 2(3y – 4) = 2×3y – 2×4 = 6y – 8
展开：2(3y – 4) = 2×3y – 2×4 = 6y – 8
5. Factorising expressions | 因式分解
Factorising is the reverse of expanding. You look for a common factor in every term and take it outside a bracket. For example, 10x + 15 can be rewritten as 5(2x + 3) because 5 is a factor of both 10 and 15. Factorising is essential for simplifying fractions and solving quadratic equations later on.
因式分解是展开的逆运算。你要找出每一项的公因数，并把它提到括号外面。例如，10x + 15 可以改写为 5(2x + 3)，因为 5 同时是 10 和 15 的因数。因式分解对于以后化简分式和解二次方程至关重要。
To factorise 6x² + 9x, the common factor is 3x, leaving 3x(2x + 3). Always check your factorisation by expanding back mentally.
分解 6x² + 9x，公因数为 3x，得到 3x(2x + 3)。每次分解后，都要在脑中反展开检查一遍。
6. Solving linear equations | 解一元一次方程
An equation shows that two expressions are equal. To solve it, you must find the value of the unknown that keeps the balance. The golden rule: whatever you do to one side, you must do to the other. For x + 5 = 12, subtract 5 from both sides to get x = 7. For 3x = 21, divide both sides by 3 to find x = 7. Always present your answer clearly.
方程表示两个表达式相等。要解方程，就必须找出使等式保持平衡的未知数的值。黄金法则是：你对等号一边做什么，对另一边也必须做相同的运算。对于 x + 5 = 12，两边同时减去 5 得 x = 7。对于 3x = 21，两边同时除以 3 得 x = 7。答案呈现要清晰明了。
Solve: 2x + 3 = 11 → 2x = 8 → x = 4
解方程：2x + 3 = 11 → 2x = 8 → x = 4
7. Equations with brackets | 带括号的方程
If an equation contains brackets, expand them first before solving. For example, 2(3x + 1) = 14 becomes 6x + 2 = 14. Then subtract 2 from both sides to get 6x = 12, and finally divide by 6 to find x = 2. This process keeps the equation balanced at every step.
如果方程中含有括号，需先展开括号再求解。例如，2(3x + 1) = 14 先展开为 6x + 2 = 14，然后两边减 2 得 6x = 12，最后除以 6 得到 x = 2。这个过程能确保方程每一步都保持平衡。
Sometimes you will have brackets on both sides. Expand them fully, then bring variable terms to one side and constants to the other.
有时括号会出现在等号两边。全部展开后，将含变量项移到一边，常数项移到另一边。
8. Equations with fractions | 含分数的方程
When an equation includes fractions, the quickest way is to multiply every term by the lowest common denominator (LCD). For x/4 + 1 = 3, multiply everything by 4 to get x + 4 = 12, so x = 8. This eliminates fractions early and makes the algebra easier to handle.
当方程中含有分数时，最快的方法是将每一项都乘以最小公分母（LCD）。例如 x/4 + 1 = 3，每项乘以 4 得 x + 4 = 12，从而 x = 8。这样可以尽早消去分数，让代数运算更容易处理。
(2x)/3 = 10 → multiply by 3 → 2x = 30 → x = 15
(2x)/3 = 10 → 两边乘3 → 2x = 30 → x = 15
9. Forming and solving equations from word problems | 应用题建立方程
Word problems ask you to translate everyday situations into algebra. Read carefully to identify the unknown and represent it with a letter. Then build an equation step by step. For instance, ‘I think of a number, double it and add 6, and the result is 20.’ Let the number be n: the equation is 2n + 6 = 20. Solve to find n = 7.
应用题要求你将日常情景翻译成代数语言。仔细阅读，找出未知数并用字母表示，然后逐步建立方程。例如，“我想一个数，把它翻倍再加6，结果是20。”设这个数为 n，方程就是 2n + 6 = 20。解方程得 n = 7。
Key words: ‘more than’ means +, ‘less than’ means -, ‘times’ or ‘product’ means ×, ‘divided by’ means ÷. Highlight these to build the equation correctly.
关键词提示：“多于”用加号，“少于”用减号，“倍”或“乘积”用乘号，“除以”用除号。找出这些词，正确构建方程。
10. Substitution and formulae | 代入与公式
Substitution means replacing letters in an expression with given numbers. If a = 3 and b = 7, then 2a + b = 2×3 + 7 = 6 + 7 = 13. You will also work with formulae such as the perimeter of a rectangle P = 2(l + w). Substitute the known lengths to find the perimeter. Always use brackets when substituting negative numbers to avoid sign errors.
代入意味着用给定的数字替换表达式中的字母。如果 a = 3，b = 7，那么 2a + b = 2×3 + 7 = 6 + 7 = 13。你还会用到像矩形周长公式 P = 2(l + w) 这样的公式，代入已知边长求周长。代入负数时一定要用括号，避免符号错误。
If x = -2, find 3x²: 3(-2)² = 3 × 4 = 12
若 x = -2，求 3x²：3(-2)² = 3 × 4 = 12
11. Checking your solutions | 检验答案
Always substitute your found value back into the original equation to check it works. If you solved 5x – 2 = 3x + 6 and got x = 4, test: left side 5×4 – 2 = 18, right side 3×4 + 6 = 18. Both sides match, so the solution is correct. This habit prevents careless mistakes and builds confidence.
始终把求出的值代回原方程，检验它是否成立。如果你解 5x – 2 = 3x + 6 得到 x = 4，检验一下：左边 5×4 – 2 = 18，右边 3×4 + 6 = 18。两边相等，解是正确的。养成这个习惯可以避免粗心错误，增强信心。
12. Common pitfalls and tips | 常见错误与提示
Many students forget to change signs when moving terms across the equals sign. Instead of ‘moving’, think ‘doing the opposite operation to both sides’. Also, watch out for missing the invisible multiplication sign between a number and a variable, like 3x meaning 3 × x. Finally, always write each step on a new line to keep your working clear and logical.
许多学生在把项从等号一边移到另一边时会忘记变号。不要想着“移项”，而应考虑“对两边做相反的运算”。此外，注意数字与变量之间隐藏的乘号，比如 3x 表示 3 × x。最后，每一步都要换行书写，使解题过程清晰、有条理。
Practise a variety of questions daily, starting with easy ones and gradually increasing difficulty. Use colours to mark like terms and underline key operations. With consistent effort, algebra will become a favourite topic rather than a hurdle.
每天练习不同类型的题目，从简单的开始，逐渐增加难度。用不同颜色标出同类项，并给关键运算画下划线。持之以恒地练习，代数就会变成你喜爱的课题，而不是一个障碍。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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