📚 Collecting Like Terms | 合并同类项
In algebra, expressions often contain several terms that can be simplified. Collecting like terms is the process of adding or subtracting terms that have exactly the same variable part.
在代数中，表达式常含有多个可以化简的项。合并同类项就是把变量部分完全相同的项相加或相减的过程。
This skill is essential for solving equations, expanding brackets, factorising, and simplifying algebraic fractions. It is one of the most frequently tested topics in IGCSE Mathematics.
这项技能对于解方程、展开括号、因式分解和化简代数分式都至关重要。它也是 IGCSE 数学中最常考查的主题之一。
1. What Are Like Terms? | 什么是同类项
A term is a single number, a variable, or a product of numbers and variables. For example, 3x, -5y, 7, and 4xy are all terms.
项是一个单独的数、一个变量，或数与变量的乘积。例如 3x、-5y、7 和 4xy 都是项。
Like terms are terms whose variable parts are exactly the same, including the same powers. For instance, 3x and -2x are like terms, but 3x and 3x² are not.
同类项是指变量部分完全相同的项，包括相同的幂次。例如 3x 和 -2x 是同类项，但 3x 和 3x² 不是同类项。
Numbers without variables, such as 5 and -8, are also like terms because they are both constants. Constants can always be combined with other constants.
没有变量的数，如 5 和 -8，也是同类项，因为它们都是常数项。常数项总可以与其他常数项合并。
| Term | 项 | Coefficient | 系数 | Variable part | 变量部分 |
|---|---|---|
| 3x | 3 | x |
| -5y | -5 | y |
| 7 | 7 | none | 无 |
| 4xy | 4 | xy |
2. The Basic Rule of Combining | 合并的基本规则
You can only add or subtract like terms. To combine them, add or subtract their coefficients and keep the variable part unchanged.
你只能对同类项进行加减。合并时，将它们的系数相加或相减，变量部分保持不变。
x = (3 + 2)x = 5x. The variable x stays the same; only the coefficients 3 and 2 are added.
例如，3x + 2x = (3 + 2)x = 5x。变量 x 保持不变；只有系数 3 和 2 相加。
3x + 2x = (3 + 2)x = 5x
nto a single term because their variable parts differ. Writing 3x + 4y = 7xy is incorrect.
不同类项，如 3x 和 4y，不能合并为一项，因为它们的变量部分不同。写 3x + 4y = 7xy 是错误的。
3. Single Variable Examples | 单变量例题
Simplify 7a + 3a - 2a.
化简 7a + 3a - 2a。
ain the variable a, so add and subtract the coefficients: 7 + 3 - 2 = 8. The answer is 8a.
这三项都含有变量 a，因此将系数相加再相减：7 + 3 - 2 = 8。答案是 8a。
7a + 3a - 2a = (7 + 3 - 2)a = 8a
Simplify 5x + 3y + 2x - y.
化简 5x + 3y + 2x - y。
rms: 5x + 2x = 7x. Collect the y terms: 3y - y = 2y. The simplified expression is 7x + 2y.
合并 x 项：5x + 2x = 7x。合并 y 项：3y - y = 2y。化简后的表达式为 7x + 2y。
5x + 3y + 2x - y = 7x + 2y
4. Powers Must Match Exactly | 幂次必须完全一致
When terms have powers, only terms with the same variable and the same exponent are like terms.
当项含有幂次时，只有变量相同且指数相同的项才是同类项。
x and x² are not like terms because the exponents are different. Therefore, 3x + 2x² cannot be simplified further.
x 和 x² 不是同类项，因为指数不同。因此 3x + 2x² 不能进一步化简。
However, 4x² and -x² are like terms, so 4x² - x² = 3x².
然而，4x² 和 -x² 是同类项，所以 4x² - x² = 3x²。
4x² - x² = 3x²
Similarly, x³ and x² are not like terms. Keep the power exactly as it is when combining coefficients.
同样，x³ 和 x² 不是同类项。合并系数时，幂次必须完全保持原样。
5. Terms with More Than One Variable | 含多个变量的项
For products of variables, the order does not matter, but the variables and their exponents must match. For example, 3xy and -5xy are like terms.
对于变量的乘积，顺序无关紧要，但变量及其指数必须相同。例如 3xy 和 -5xy 是同类项。
However, 3xy and 3x are not like terms because one has y and the other does not. Also, 3xy and 3x²y are not like terms because the powers of x differ.
然而，3xy 和 3x 不是同类项，因为一个含有 y，另一个没有。此外，3xy 和 3x²y 也不是同类项，因为 x 的幂次不同。
Simplify 2ab + 3a + 4ab - a.
化简 2ab + 3a + 4ab - a。
t the ab terms: 2ab + 4ab = 6ab. Collect the a terms: 3a - a = 2a. The result is 6ab + 2a.
合并 ab 项：2ab + 4ab = 6ab。合并 a 项：3a - a = 2a。结果为 6ab + 2a。
2ab + 3a + 4ab - a = 6ab + 2a
6. Negative Coefficients and Subtraction | 负系数与减法
Subtracting a term is the same as adding its negative. Take care with signs when combining coefficients.
减去一项等同于加上它的相反数。合并系数时要特别注意符号。
Simplify 6x - 10x + 3.
化简 6x - 10x + 3。
The coefficient calculation is 6 - 10 = -4, so the expression becomes -4x + 3.
系数计算为 6 - 10 = -4，因此表达式变为 -4x + 3。
6x - 10x + 3 = -4x + 3
Simplify 2p - 5q - 3p + q.
化简 2p - 5q - 3p + q。
For p: 2p - 3p = -p. For q: -5q + q = -4q. The simplified expression is -p - 4q.
对于 p：2p - 3p = -p。对于 q：-5q + q = -4q。化简后的表达式为 -p - 4q。
2p - 5q - 3p + q = -p - 4q
Remember that -p means -1p. When a variable appears without a number, its coefficient is 1 or -1.
请记住，-p 表示 -1p。当变量前没有数字时，它的系数是 1 或 -1。
7. Removing Brackets Before Collecting | 先去括号再合并
Sometimes expressions include brackets. Expand or remove the brackets first, then collect like terms.
有些表达式含有括号。先展开或去掉括号，再合并同类项。
Simplify 3(x + 2) + 2x - 1.
化简 3(x + 2) + 2x - 1。
2x - 1. Now collect x terms: 3x + 2x = 5x, and constants: 6 - 1 = 5. The answer is 5x + 5.
展开：3x + 6 + 2x - 1。现在合并 x 项：3x + 2x = 5x，常数项：6 - 1 = 5。答案为 5x + 5。
3(x + 2) + 2x - 1 = 5x + 5
gns of the terms inside when removing it. For example, 5 - (2x - 3) = 5 - 2x + 3 = 8 - 2x.
如果括号前是减号，去括号时要改变括号内各项的符号。例如 5 - (2x -
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图