📚 Algebra Essentials: Simplifying Expressions and Solving Linear Equations | 代数基础：化简表达式与解一元一次方程
Algebra is the branch of mathematics where letters and symbols are used to represent numbers. In KS3 Cambridge Mathematics, you will learn how to simplify algebraic expressions, expand brackets, and solve linear equations step by step. These skills form the foundation for almost every topic later in IGCSE and beyond, so it is important to master them now. This guide covers the key ideas you need, with clear examples and common mistakes to avoid.
代数是数学中运用字母和符号表示数字的一个重要分支。在剑桥初中数学（KS3）阶段，你将学习如何化简代数表达式、展开括号以及分步解一元一次方程。这些技能是后续 IGCSE 乃至更高水平数学中几乎所有专题的基础，因此现在打好基础非常重要。本指南将带你梳理核心概念，并通过清晰的例子和常见错误提醒帮助你掌握这些内容。
1. Understanding Algebraic Terms | 理解代数项
An algebraic expression is made up of terms. A term can be a number, a letter, or a combination of numbers and letters multiplied together. For example, in the expression 5x + 3y − 7, there are three terms: 5x, 3y, and −7. The number part of a term is called the coefficient, so in 5x the coefficient is 5. A term that contains no variable, such as −7, is called a constant.
代数表达式由项组成。项可以是一个数、一个字母，或者数与字母的乘积。例如，在表达式 5x + 3y − 7 中，有三个项：5x、3y 和 −7。项中的数字部分叫作系数，因此 5x 的系数是 5。不含变量的项（如 −7）叫作常数项。
| Term | Coefficient | Variable |
|---|---|---|
| 6x | 6 | x |
| −4y | −4 | y |
| 9 | — | constant |
it means 2 × a × b. Letters next to each other or next to numbers are multiplied together.
理解代数的基本词汇有助于你正确读写表达式。当看到一个像 2ab 这样的项时，它表示 2 × a × b。字母彼此相邻或字母与数字相邻时，都表示相乘。
2. Collecting Like Terms | 合并同类项
Like terms are terms that contain exactly the same variable or variables raised to the same power. For example, 3x and 5x are like terms, but 3x and 3x² are not. When simplifying an expression, you can only add or subtract like terms. To collect like terms, add or subtract their coefficients while keeping the variable part unchanged.
同类项是指含有完全相同变量且变量次数相同的项。例如，3x 和 5x 是同类项，但 3x 和 3x² 不是。化简表达式时，只能对同类项进行加减。合并同类项时，将它们的系数相加或相减，变量部分保持不变。
4x + 7x − 2x = (4 + 7 − 2)x = 9x
This rule works because the variable x represents the same unknown value in each term. Think of it like grouping objects: 4 apples plus 7 apples minus 2 apples gives 9 apples. The same logic applies when the terms include more than one variable.
这一规则之所以成立，是因为变量 x 在每个项中代表相同的未知数。可以把它想象成物品分组：4 个苹果加 7 个苹果再减去 2 个苹果，得到 9 个苹果。当项中包含多个变量时，逻辑同样适用。
3a + 5b + 2a − b = 5a + 4b
Notice that 3a and 2a were combined to give 5a, while 5b and −b were combined to give 4b. You cannot combine the a terms with the b terms because they represent different unknowns.
注意，3a 和 2a 合并得到 5a，而 5b 和 −b 合并得到 4b。不能把含 a 的项与含 b 的项合并，因为它们代表不同的未知数。
3. Expanding Brackets | 展开括号
+ c) = ab + ac. For example, to expand 3(x + 4), multiply 3 by x and then multiply 3 by 4.
展开括号就是用括号外的项乘括号内的每一项。这基于分配律：a(b + c) = ab + ac。例如，展开 3(x + 4)，就是用 3 乘 x，再用 3 乘 4。
3(x + 4) = 3 × x + 3 × 4 = 3x + 12
When there is a negative sign outside the bracket, you must be careful to multiply every term inside by the negative number. For instance, −2(x − 5) becomes −2x + 10, because −2 multiplied by −5 gives +10.
当括号外是负号时，必须小心地将括号内的每一项都乘以这个负数。例如，−2(x − 5) 展开后为 −2x + 10，因为 −2 乘以 −5 等于 +10。
Expanding brackets is often a necessary first step before solving an equation. If you expand incorrectly, the final answer will be wrong even if your method is correct. Always double-check the signs.
展开括号通常是解方程前的必要步骤。如果展开错误，即使方法正确，最终答案也会出错。务必仔细检查符号。
4. What Is an Equation? | 什么是方程？
ans finding the value of the unknown that makes the statement true. Unlike an expression, an equation can be solved to find a specific number.
方程是两个表达式相等的数学陈述。它包含等号，例如 x + 3 = 7 或 2x − 5 = 11。解方程就是找出使等式成立的未知数的值。与表达式不同，方程可以求出具体的数值。
The key principle for solving equations is maintaining balance. You can think of an equation as a balanced set of scales. Whatever you do to one side of the equation, you must do to the other side to keep the scales balanced. This is called the balance method.
解方程的关键原则是保持平衡。你可以把方程想象成一个平衡的天平。无论对方程的一边做什么运算，对另一边也必须做同样的运算，才能保持天平平衡。这就是平衡法。
5. Solving One-Step Equations | 解一步方程
olate x. This gives x = 7. Similarly, in x − 4 = 9, you add 4 to both sides to get x = 13.
一步方程只需进行一次运算就能求出解。例如，在 x + 5 = 12 中，你需要从等号两边同时减去 5 来孤立 x，得到 x = 7。类似地，在 x − 4 = 9 中，两边同时加 4，得到 x = 13。
x + 5 = 12
x + 5 − 5 = 12 − 5
x = 7
7. If x ÷ 5 = 4, multiply both sides by 5 to get x = 20. Always show your working clearly.
对于乘法和除法方程，使用逆运算求解。如果 3x = 21，两边同时除以 3，得到 x = 7。如果 x ÷ 5 = 4，两边同时乘以 5，得到 x = 20。解题过程一定要书写清楚。
6. Solving Two-Step Equations | 解两步方程
e variable x is first multiplied by 2 and then 3 is added. To solve it, reverse the order of operations: undo the addition or subtraction first, then undo the multiplication or division.
两步方程包含两种运算。例如，在 2x + 3 = 11 中，变量 x 先乘以 2，再加 3。解这类方程时，要逆向运算：先消除加或减，再消除乘或除。
2x + 3 = 11
2x + 3 − 3 = 11 − 3
2x = 8
x = 4
This reverse order is important. Many students mistakenly divide by 2 before subtracting 3, which leads to an incorrect result. Always write each line of working to make your method clear and to reduce errors.
这种逆向顺序非常重要。许多学生错误地先除以 2 再减 3，导致结果错误。务必一步一步写出解题过程，使方法清晰并减少错误。
. First add 2 to both sides to get x ÷ 4 = 5, then multiply both sides by 4 to get x = 20.
另一个例子涉及减法和除法：(x ÷ 4) − 2 = 3。首先两边同时加 2，得到 x ÷ 4 = 5，然后两边同时乘以 4，得到 x = 20。
7. Equations with Brackets | 含括号的方程
de if possible, and then solve using the balance method. For example, solve 3(x + 2) = 21.
当方程含有括号时，首先用分配律展开括号。展开后，如果可能，先化简两边，再用平衡法求解。例如，解方程 3(x + 2) = 21。
3(x + 2) = 21
3x + 6 = 21
3x = 15
x = 5
Expanding first makes the equation easier to handle. If you try to divide both sides by 3 before expanding, you could also solve it, but many students make sign errors when brackets are involved. Expanding is generally the safer first step.
先展开括号会使方程更易于处理。如果你尝试在展开前先除以 3，也可以求解，但涉及括号时很多学生容易犯符号错误。一般来说，先展开是更稳妥的步骤。
8. Variables on Both Sides | 变量在等号两边
ect the variable terms on one side and the constant terms on the other. You can do this by adding or subtracting the same term from both sides.
有些方程的未知数出现在等号两边，例如 5x + 2 = 3x + 10。解这类方程时，先把变量项集中到一边，把常数项集中到另一边。可以通过在两边同时加上或减去相同的项来实现。
5x + 2 = 3x + 10
5x − 3x + 2 = 3x − 3x + 10
2x + 2 = 10
2x = 8
x = 4
It does not matter which side you choose for the variable, as long as you perform the same operation on both sides. Some students prefer to keep the variable positive, so they subtract the smaller variable term from both sides.
选择把变量放在哪一边都可以，只要对两边执行相同的运算。有些学生喜欢让变量的系数为正，因此他们会从两边减去较小的变量项。
9. Checking Your Answer | 验证答案
ace x with 4: 2 × 4 + 3 = 8 + 3 = 11. Since both sides are equal, the solution is correct.
解完方程后，应始终将答案代回原方程进行验证。例如，如果你求出方程 2x + 3 = 11 的解为 x = 4，将 x 替换为 4：2 × 4 + 3 = 8 + 3 = 11。两边相等，说明解正确。
Checking is especially useful in exams because it helps you catch small arithmetic mistakes. It takes only a few seconds but can save you several marks. Make this a habit for every equation you solve.
验证在考试中特别有用，因为它能帮助你发现小型的计算错误。这只需要几秒钟，却能帮你避免丢分。养成每解一个方程都验证的好习惯。
10. Common Errors and Exam Tips | 常见错误与考试技巧
peration you do to one side must be done to the other; adding or subtracting does not simply move a term without changing its sign.
一个常见错误是在把项移到等号另一边时忘记变号。例如，在 2x + 3 = 9 中，有些学生写成 2x = 9 + 3，而不是 2x = 9 − 3。记住，对一边做什么运算，对另一边也要做同样的运算；加减运算并不是简单地把项移过去而不变号。
Another common error is combining unlike terms, such as writing 3x + 2 as 5x. This is incorrect because 3x and 2 are not like terms. Only terms with the same variable can be added or subtracted. Also, be careful when expanding brackets with negative signs.
另一个常见错误是合并不是同类项的项，例如把 3x + 2 写成 5x。这是错误的，因为 3x 和 2 不是同类项。只有含相同变量的项才能相加或相减。此外，展开带负号的括号时要格外小心。
- Write each step on a new line to make your method clear — 每一步另起一行书写，使解题过程清晰。
- Check signs carefully when expanding brackets — 展开括号时仔细检查符号。
- Always substitute your answer to verify it satisfies the original equation — 总是代入答案验证是否满足原方程。
- If the variable disappears and you get a false statement, the equation has no solution — 如果变量消去后得到错误等式，则方程无解。
By avoiding these errors and following the balance method, you can solve KS3 linear equations confidently and accurately.
避免这些错误并遵循平衡法，你就能自信而准确地解答 KS3 一元一次方程。
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