📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, algebraic expressions can often be made much shorter by collecting like terms. This skill is fundamental because it appears when solving equations, expanding brackets, factorising and simplifying formulae. Once you can recognise terms that may be added or subtracted, algebra becomes a much more manageable process.
在 IGCSE 数学中，代数表达式通常可以通过合并同类项来大大缩短。这项技能是基础，因为在解方程、去括号、因式分解和化简公式时都会用到它。一旦你能识别哪些项可以相加或相减，代数就会变得更加容易掌握。
This guide explains what terms are, how to identify like terms, and how to simplify expressions step by step. It also highlights common errors and includes IGCSE-style practice questions.
本指南解释什么是项、如何识别同类项，以及如何逐步化简表达式。它还会指出常见错误，并提供 IGCSE 风格练习题。
1. Terms and Coefficients | 项与系数
An algebraic expression is made up of terms separated by plus or minus signs. In the expression 7x + 3y - 5, there are three terms: 7x, 3y and -5. The number part in a term such as 7x is called the coefficient; here 7 is the coefficient and x is the variable.
代数表达式由用加号或减号分隔的项组成。在表达式 7x + 3y - 5 中，有三项：7x、3y 和 -5。诸如 7x 这样的项中的数字部分称为系数；这里 7 是系数，x 是变量。
A constant term is a number on its own, like -5. It has no variable part. When simplifying, constant terms are collected together separately from variable terms.
常数项是单独出现的数字，例如 -5。它没有变量部分。化简时，常数项要与变量项分开合并。
2. What Are Like Terms? | 什么是同类项
Like terms have exactly the same variable factors raised to the same powers. For example, 4x and -9x are like terms because both contain x raised to the power 1. However, 4x and 4x² are not like terms because the powers of x are different.
同类项具有完全相同的变量因式且每个变量的次数相同。例如，4x 和 -9x 是同类项，因为它们都含有一次幂的 x。然而，4x 和 4x² 不是同类项，因为 x 的次数不同。
Similarly, 3xy and -2xy are like terms, but 3xy and 3x²y are not like terms. For multi-variable terms, every variable and every power must match exactly.
类似地，3xy 和 -2xy 是同类项，但 3xy 和 3x²y 不是同类项。对于多变量项，每个变量及其次数都必须完全一致。
3. The Basic Rule of Collecting | 合并同类项的基本规则
t unchanged. For example, 5x + 3x means (5 + 3)x, which gives 8x. Likewise, 10y - 6y = 4y.
合并同类项时，只将系数相加或相减，变量部分保持不变。例如，5x + 3x 表示 (5 + 3)x，得到 8x。同样，10y - 6y = 4y。
each term. In 10y - 6y, the second coefficient is -6, so the calculation is 10 + (-6) = 4.
始终保留每项前面系数的正负号。在 10y - 6y 中，第二项的系数是 -6，因此计算为 10 + (-6) = 4。
4. Single Variable Expressions | 单变量表达式
nswer is 10a. For mixed expressions, collect variable terms and constant terms separately.
化简 12a - 5a + 3a。合并系数：12 - 5 + 3 = 10，因此答案为 10a。对于混合表达式，分别合并变量项和常数项。
4 = (7b - 2b) + (9 + 4) = 5b + 13. The variable terms give 5b, and the constants give 13.
例如：7b + 9 - 2b + 4 = (7b - 2b) + (9 + 4) = 5b + 13。变量项得到 5b，常数项得到 13。
5. Expressions with Powers | 含幂的表达式
x² + 3x² = 4x², but x² + x cannot be simplified further because the powers are different.
当项含有幂时，只有变量完全相同且次数完全相同的项才能合并。例如，x² + 3x² = 4x²，但 x² + x 不能继续化简，因为次数不同。
5x² - 2x + 3x² + 7x = 8x² + 5x
rms: -2x + 7x = 5x. The final answer is written with the higher power first by convention.
在这个例子中，合并 x² 项：5x² + 3x² = 8x²。然后合并 x 项：-2x + 7x = 5x。按照惯例，最终答案将次数较高的项写在前。
Never combine different powers, such as x³ and x². They are unlike terms and must remain separate in the simplified answer.
切勿合并不同次数的项，例如 x³ 和 x²。它们不是同类项，在化简后的答案中必须保持分开。
6. Multi-Variable Terms | 多变量项
For terms with more than one variable, all variable letters and their powers must match exactly. For example, 2xy and -5xy are like terms, so they combine to -3xy. But 4xy and 4x²y are unlike terms.
对于含有多个变量的项，所有变量字母及其次数必须完全一致。例如，2xy 和 -5xy 是同类项，因此合并为 -3xy。但 4xy 和 4x²y 不是同类项。
Consider 3x²y and 3xy². Although they contain the same letters, the powers are attached to different variables, so these terms cannot be combined.
考虑 3x²y 和 3xy²。虽然它们含有相同的字母，但次数分配在不同的变量上，因此这些项不能合并。
Example: 6xy + 2x - 3xy + 5x = (6xy - 3xy) + (2x + 5x) = 3xy + 7x.
例子：6xy + 2x - 3xy + 5x = (6xy - 3xy) + (2x + 5x) = 3xy + 7x。
7. Removing Brackets First | 先去括号再合并
Many IGCSE questions ask you to simplify expressions that include brackets. Expand the brackets first, then collect like terms. For example: 3(x + 2y) - 2(x - y).
许多 IGCSE 题目要求化简含有括号的表达式。先去括号，再合并同类项。例如：3(x + 2y) - 2(x - y)。
Expand to get 3x + 6y - 2x + 2y. Then collect the x terms: 3x - 2x =
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