📚 Collecting Like Terms in IGCSE Algebra | IGCSE 代数中的合并同类项
In IGCSE algebra, collecting like terms is one of the first and most important simplification skills. It allows you to rewrite an expression such as 3x + 5x as 8x, making later work with equations, inequalities and graphs much easier.
在 IGCSE 代数中，合并同类项是最先学习也最重要的化简技能之一。它能把 3x + 5x 这样的式子写成 8x，让后续的方程、不等式和图像问题简单得多。
1. What Are "Like Terms"? | 什么是 "同类项"
Like terms are terms that contain exactly the same variable or variables raised to exactly the same powers. Only the number in front, called the coefficient, can be different.
同类项是指含有完全相同的变量、且各变量次数完全相同的项。只有前面的数，也就是系数，可以不同。
For example, 4x and -7x are like terms because both contain x to the power 1. The terms 5x² and 3x² are also like terms because both contain x².
例如，4x 与 -7x 是同类项，因为它们都含有 x 的 1 次方。5x² 与 3x² 也是同类项，因为它们都含有 x²。
| Terms | Like or Unlike? | Reason |
|---|---|---|
| 4x and -7x | Like | Same variable x, same power 1 |
| 5x² and 3x² | Like | Same variable x, same power 2 |
| 4x and 4x² | Unlike | Different powers of x |
The table shows that only the variable part, including its power, decides whether terms can be combined.
上表说明，只有变量部分，包括它的次数，决定这些项能否合并。
2. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient for a moment and look only at the variable part, including its power.
识别同类项时，暂时忽略系数，只看变量部分，包括它的次数。
The terms 7a, -2a and 0.5a are like terms because they all contain exactly a to the power 1. The numbers 7, -2 and 0.5 are simply coefficients.
7a、-2a 和 0.5a 是同类项，因为它们都只含有 a 的 1 次方。数字 7、-2 和 0.5 只是系数。
For multi-variable terms, check every variable and every power. The terms 3xy and 5xy are like terms, but 3xy and 3x²y are not.
对于多变量项，要检查每一个变量和每一次方。3xy 和 5xy 是同类项，但 3xy 和 3x²y 不是。
3. The Golden Rule: Only Add or Subtract the Coefficients | 黄金法则：只加减系数
When you collect like terms, the variable part does not change. You simply add or subtract the coefficients.
合并同类项时，变量部分不变。你只需要把系数相加或相减。
3x + 5x = (3 + 5)x = 8x
object. If you have 3 apples and add 5 apples, you get 8 apples. Similarly, 3x + 5x = 8x.
可以把 x 想成一种物品。如果你有 3 个苹果，再加 5 个苹果，就有 8 个苹果。同样，3x + 5x = 8x。
For subtraction, the same rule applies.
减法也遵循同样的规则。
9y - 4y = (9 - 4)y = 5y
4. Single-Variable Examples | 单变量示例
Work through each expression by circling or underlining groups of like terms before simplifying.
化简前，先圈出或划出每组同类项，再逐组处理。
2x + 7x - 3x = (2 + 7 - 3)x = 6x
Another example with mixed powers is shown below.
下面是一个含有不同次数的例子。
4x + 3x² + 6x - x²
+ 6x = 10x. Collect the x² terms: 3x² - x² = 2x². The simplified expression is 10x + 2x².
合并 x 项：4x + 6x = 10x。合并 x² 项：3x² - x² = 2x²。化简结果是 10x + 2x²。
5. Multi-Variable Terms | 含多个变量的项
For terms with more than one variable, all variables and all powers must match exactly. For example, 3ab and -5ab are like terms, but 3ab and 3a²b are not.
对于含有多个变量的项，所有变量及次数必须完全一致。例如，3ab 与 -5ab 是同类项，但 3ab 与 3a²b 不是。
Simplify 2xy + 5x + 3xy - 7x.
化简 2xy + 5x + 3xy - 7x。
The xy terms give 2xy + 3xy = 5xy. The x terms give 5x - 7x = -2x. The result is 5xy - 2x.
xy 项是 2xy + 3xy = 5xy。x 项是 5x - 7x = -2x。结果是 5xy - 2x。
Notice that 5xy and -2x cannot be combined because one contains xy and the other contains only x.
注意 5xy 和 -2x 不能合并，因为一个含有 xy，另一个只含有 x。
6. Constants and Mixed Expressions | 常数项与混合表达式
. They are like terms with each other. For example, 6 and -2 are like terms, so 6 - 2 = 4.
常数项是不含变量的数字，它们彼此是同类项。例如，6 与 -2 是同类项，因此 6 - 2 = 4。
+ 3 + 2a - 8. The a terms: 5a + 2a = 7a. The constants: 3 - 8 = -5. The answer is 7a - 5.
化简 5a + 3 + 2a - 8。a 项：5a + 2a = 7a。常数项：3 - 8 = -5。答案是 7a - 5。
Next, simplify 4b + 7 - 2b + 3 + b. The b terms: 4b - 2b + b = 3b. The constants: 7 +
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