📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, collecting like terms is the first essential skill for simplifying expressions. This article explains how to identify, combine, and check like terms so you can handle harder algebra with confidence.
在 IGCSE 代数中，合并同类项是化简表达式的首要基本技能。本文将讲解如何识别、合并和检查同类项，帮助你自信地应对更复杂的代数题。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that contain exactly the same variable letters and the same powers. For example, 3x and 5x are like terms because both contain x to the power 1. However, 3x and 3x² are not like terms because the powers differ.
同类项是指含有完全相同的字母变量且各字母的指数也相同的项。例如，3x 和 5x 是同类项，因为两者都含有 x 的一次方。但 3x 和 3x² 不是同类项，因为幂次不同。
Only the coefficients may be different. Terms such as 4xy, -2xy, and 7xy are all like terms because the variable part xy is identical.
只有系数可以不同。像 4xy、-2xy 和 7xy 都是同类项，因为它们的变量部分 xy 完全相同。
2x, −7x, 0.5x are like terms
2x, 2x², 2y are not like terms
2. Identifying Like Terms in Expressions | 在表达式中识别同类项
To identify like terms in a longer expression, ignore the numbers in front and focus on the variable part. In 6a + 3b - 2a + 5b, the a terms are 6a and -2a, and the b terms are 3b and 5b.
在较长表达式中识别同类项时，先忽略数字系数，关注变量部分。在 6a + 3b - 2a + 5b 中，a 项是 6a 和 -2a，b 项是 3b 和 5b。
A useful method is to underline or circle each set of like terms with the same mark. This helps prevent missing terms when signs are involved.
一个有用的方法是用相同的标记对每组同类项画线或画圈。当含有符号时，这样可以防止遗漏项。
| Expression 表达式 | Like terms 同类项 |
|---|---|
| 6a + 3b - 2a + 5b | 6a and -2a; 3b and 5b |
| 4x² + 3x - x² + 2x | 4x² and -x²; 3x and 2x |
6a + 3b − 2a + 5b = 4a + 8b
3. The Basic Rule of Combining | 合并的基本法则
efficients and keep the variable part unchanged. For example, 7m + 2m means (7 + 2)m = 9m.
合并同类项时，只对系数进行加减运算，变量部分保持不变。例如，7m + 2m 表示 (7 + 2)m = 9m。
iable part represents a common unit. Just as 7 apples + 2 apples = 9 apples, 7m + 2m = 9m.
这是因为变量部分代表一个共同单位。就像 7 个苹果加 2 个苹果等于 9 个苹果，7m + 2m = 9m。
ax + bx = (a + b)x
The same rule applies to subtraction: the signs of the coefficients are included in the addition.
同样的法则也适用于减法：系数的符号包含在加法运算中。
4. Adding and Subtracting Like Terms | 同类项的加法与减法
e coefficient rule. For 10p - 3p, subtract the coefficients: 10 - 3 = 7, so 10p - 3p = 7p.
加法和减法遵循相同的系数法则。对于 10p - 3p，系数相减：10 - 3 = 7，因此 10p - 3p = 7p。
If the result has a coefficient of 1, write the variable alone: 8k - 7k = k, not 1k.
如果结果系数为 1，只需写出变量本身：8k - 7k = k，而不是 1k。
If the result is zero, write 0: 5n - 5n = 0.
如果结果为零，写 0：5n - 5n = 0。
10p − 3p = 7p
8k − 7k = k
5. Constants and Terms Without Variables | 常数项与不含变量的项
Constants are terms without variables, such as 4, -9, and 12. They are like terms with each other and can be combined separately from variable terms.
常数项是没有变量的项，例如 4、-9 和 12。它们彼此是同类项，可以与含变量的项分开合并。
combine the x terms and constants separately: 3x + 2x = 5x and 5 - 7 = -2, giving 5x - 2.
在 3x + 5 + 2x - 7 中，分别合并 x 项和常数项：3x + 2x = 5x，5 - 7 = -2，结果为 5x - 2。
3x + 5 + 2x − 7 = 5x − 2
Always keep the final answer tidy: variable terms usually come first, followed by the constant term.
最后答案要保持整洁：通常先写含变量的项，再写常数项。
6. Working with Negative Coefficients | 处理负系数
ngs to that term. For 4y - 9y, think of -9y as negative nine y, so 4 - 9 = -5, giving -5y.
当负号出现在某项之前时，它属于该项。对于 4y - 9y，把 -9y 看作负九 y，因此 4 - 9 = -5，结果为 -5y。
In longer expressions, keep the sign with each term. For 2x - 5 + 3x - 4, the x coefficients are 2 and 3, while the constants are -5 and -4.
在较长表达式中，符号要跟随每一项。对于 2x - 5 + 3x - 4，x 的系数是 2 和 3，常数项是 -5 和 -4。
2x − 5 + 3x − 4 = 5x − 9
ally careful when subtracting a term with a negative coefficient, such as 6a - (-2a) = 8a.
减去一个负系数的项时要格外小心，例如 6a - (-2a) = 8a。
7. Expressions with Multiple Variables | 含有多个变量的表达式
Expressions can contain more than one variable. Terms must have the same combination of letters to be combined. In 2xy + 3x + 5xy - x, the xy terms combine to 7xy and the x terms combine to 2x.
表达式可以含有多个变量。只有字母组合完全相同的项才能合并。在 2xy + 3x + 5xy - x 中，xy 项合并为 7xy，x 项合并为 2x。
Be careful: xy and x are not like terms because xy has an extra variable y.
注意：xy 和 x 不是同类项，因为 xy 多了一个变量 y。
2xy + 3x + 5xy − x = 7xy + 2x
Similarly, ab and a²b are not like terms because the power of a is different.
类似地，ab 和 a²b 不是同类项，因为 a 的指数不同。
8. Powers and Like Terms | 幂与同类项
Terms with the same variable but different powers are not like terms. For example, x and x² cannot be combined because x is x¹ and x² has power 2.
变量相同但幂次不同的项不是同类项。例如，x 和 x² 不能合并，因为 x 是 x¹，而 x² 的幂次为 2。
terms because both contain x². Their coefficients combine: 5 - 3 = 2, so 5x² - 3x² = 2x².
像 5x² 和 -3x² 这样的项是同类项，因为两者都含有 x²。它们的系数合并：5 - 3 = 2，因此 5x² - 3x² = 2x²。
Remember: 2x + 3x² cannot be simplified further. The answer remains 2x + 3x².
请记住：2x + 3x² 不能进一步化简。答案仍为 2x + 3x²。
5x² − 3x² = 2x²
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