📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, algebra often looks like a string of symbols, numbers and letters. One of the first skills you need is to simplify expressions by collecting like terms. This makes expressions shorter, clearer and ready for solving equations or drawing graphs.
在 IGCSE 数学中，代数常常看起来像一串符号、数字和字母。你需要掌握的第一项技能就是通过合并同类项来化简表达式。这样能让表达式更短、更清晰，也为解方程或画图像做好准备。
1. What Are Terms in Algebra? | 代数中的项是什么
A term is a single mathematical part of an expression. Terms are separated by + or − signs. For example, in 5x + 3y − 7, the terms are 5x, 3y and 7. A term can be a number, a variable, or a product of numbers and variables.
项是表达式中的一个独立数学部分。项之间用 + 或 − 号分隔。例如，在 5x + 3y − 7 中，项是 5x、3y 和 7。项可以是一个数、一个变量，或者是数与变量的乘积。
Identifying terms correctly is essential before simplifying. The sign in front of a term belongs to that term. In the expression 4a − 2b + 6, the three terms are 4a, −2b and 6.
在化简之前，正确识别项非常重要。项前面的符号属于该项。在表达式 4a − 2b + 6 中，三个项分别是 4a、−2b 和 6。
2. Coefficients, Variables and Constants | 系数、变量与常数
A variable is a letter that stands for an unknown or changing value, such as x, y or n. A coefficient is the number multiplying the variable. In 7x, 7 is the coefficient and x is the variable. A constant is a term with no variable, such as 4 or −9.
变量是代表未知数或变化值的字母，例如 x、y 或 n。系数是乘以变量的数。在 7x 中，7 是系数，x 是变量。常数是没有变量的项，例如 4 或 −9。
Knowing these terms helps you decide which parts of an expression can be combined. Only terms with exactly the same variable part can be collected. Constants can always be collected with other constants.
了解这些术语有助于判断表达式中的哪些部分可以合并。只有变量部分完全相同的项才能合并。常数项之间总是可以合并。
3. Defining Like Terms | 什么是同类项
Like terms have identical variable parts, including the same letters raised to the same powers. The coefficients can be different. For example, 3x and −5x are like terms, but 3x and 3x² are not like terms because the powers of x are different.
同类项具有完全相同的变量部分，包括相同字母和相同指数。系数可以不同。例如，3x 和 −5x 是同类项，但 3x 和 3x² 不是同类项，因为 x 的指数不同。
Similarly, 2ab and −7ab are like terms, while 2ab and 2a are not. Constant terms such as 6 and −11 are also like terms with each other.
类似地，2ab 和 −7ab 是同类项，而 2ab 和 2a 不是。常数项如 6 和 −11 也互为同类项。
| Like terms | 4x and 9x; 2y² and −5y²; 3ab and 6ab |
| Not like terms | 4x and 4x²; 2ab and 2a; 5x and 5y |
同类项示例：4x 和 9x、2y² 和 −5y²、3ab 和 6ab。非同类项示例：4x 和 4x²、2ab 和 2a、5x 和 5y。
4. Why Collect Like Terms? | 为什么要合并同类项
Collecting like terms simplifies an expression. Instead of writing x + x + x + x, we write 4x. This reduces mistakes and shows the structure of the expression. It is especially useful before substituting values, solving equations, or factorising.
合并同类项可以化简表达式。与其写 x + x + x + x，不如写成 4x。这会减少错误并展示表达式的结构。在代入数值、解方程或因式分解之前，这尤其有用。
In IGCSE exam questions, you often see instructions such as 'Simplify fully'. Collecting like terms is usually one of the required steps. If an expression is not simplified, you may lose accuracy marks.
在 IGCSE 考试题中，你常会看到 '完全化简' 的要求。合并同类项通常是必要步骤之一。如果表达式没有化简，你可能会失去答案准确度分。
5. Adding and Subtracting Like Terms | 合并同类项的基本加减
unchanged. For example, 5x + 3x = (5 + 3)x = 8x. For subtraction, 9y − 4y = (9 − 4)y = 5y.
要合并同类项，就将它们的系数相加或相减，变量部分保持不变。例如，5x + 3x = (5 + 3)x = 8x。对于减法，9y − 4y = (9 − 4)y = 5y。
5x + 3x = 8x
areful with signs. For example, 4x − 7x = (4 − 7)x = −3x. Also, −2a + 5a = (−2 + 5)a = 3a.
当项有负系数时，要注意符号。例如，4x − 7x = (4 − 7)x = −3x。另外，−2a + 5a = (−2 + 5)a = 3a。
You cannot combine unlike terms by addition. For example, 3x + 4y stays as 3x + 4y because the variable parts differ. The expression can be rearranged but not simplified further by addition.
不能通过加法合并非同类项。例如，3x + 4y 保持不变，因为变量部分不同。表达式可以重新排列，但不能通过加法进一步化简。
6. Collecting Terms with Powers | 含幂的同类项合并
e terms: x² + 5x² = 6x². However, x² and x are not like terms because their powers differ.
含幂的项只有在底数和指数都相同时才是同类项。例如，x² 和 5x² 是同类项：x² + 5x² = 6x²。但 x² 和 x 不是同类项，因为它们的指数不同。
d 2x² cannot be combined. Always check the exponent before adding or subtracting. Treat x and x² as completely different variable parts.
同样，2x³ − 7x³ = −5x³，但 2x³ 和 2x² 不能合并。在加减之前一定要检查指数。要把 x 和 x² 视为完全不同的变量部分。
the x² terms: 4x² − x² = 3x². Collect the x terms: 3x + 7x = 10x. Final answer: 3x² + 10x.
例如，化简 4x² + 3x − x² + 7x。合并 x² 项：4x² − x² = 3x²。合并 x 项：3x + 7x = 10x。最终答案：3x² + 10x。
7. Multi-Variable Like Terms | 多变量同类项
− 2xy = 2xy. But 4xy and 4x are not like terms, and xy and x²y are not like terms either.
对于含多个变量的项，字母及其指数必须完全一致。例如，4xy 和 −2xy 是同类项：4xy − 2xy = 2xy。但 4xy 和 4x 不是同类项，xy 和 x²y 也不是同类项。
Order of letters does not change the term, so 3ab and 5ba are like terms because multiplication is commutative. It is often helpful to write variables in alphabetical order to make like terms easier to spot.
字母顺序不会改变项，因此 3ab 和 5ba 是同类项，因为乘法满足交换律。通常把变量按字母顺序书写会更容易发现同类项。
fy 6mn + 3m − 4nm + 2m. Rewrite 4nm as 4mn. Then 6mn − 4mn = 2mn and 3m + 2m = 5m. Answer:
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