Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Welcome to this revision guide for G-2, Topic 15: Collecting Like Terms. This skill is one of the most important foundations in IGCSE Mathematics.

欢迎阅读 G-2 课本主题 15「合并同类项」复习指南。这是 IGCSE 数学中最重要的基础技能之一。

Collecting like terms means simplifying an algebraic expression by adding or subtracting terms that have exactly the same variable part.

合并同类项，就是把代数表达式中变量部分完全相同的项通过加法或减法合并，从而化简表达式。

1. What Are Like Terms? | 什么是同类项

In algebra, a term is a single number, a variable, or a product of numbers and variables. For example, 3x, -5y² and 7 are all terms.

在代数中，项是一个单独的数、一个变量，或数字与变量的乘积。例如，3x、-5y² 和 7 都是项。

Like terms are terms that have exactly the same variables, with the same powers. Their coefficients may be different.

同类项是指变量完全相同，且每个变量的次数也完全相同的项。同类项的系数可以不同。

• 3x and -9x are like terms because both contain x to the power 1.

  3x 与 -9x 是同类项，因为它们都含有一次变量 x。

• 2x² and 5x² are like terms because both contain x².

  2x² 与 5x² 是同类项，因为它们都含有 x²。

• 3x and 3x² are not like terms, because the powers of x are different.

  3x 与 3x² 不是同类项，因为 x 的次数不同。

• 4xy and 2yx are like terms, because xy = yx by the commutative property.

  4xy 与 2yx 是同类项，因为根据乘法交换律 xy = yx。

2. Why Do We Combine Like Terms? | 为什么要合并同类项

Combining like terms makes an expression shorter and easier to use. It also helps us solve equations later.

合并同类项能让表达式更简洁、更容易使用，也方便我们后续解方程。

If you have 3 apples and 2 apples, you have 5 apples. Similarly, 3x + 2x = 5x. But 3 apples and 2 bananas cannot be counted as one type, just as 3x + 2y cannot be simplified.

如果你有 3 个苹果和 2 个苹果，你一共有 5 个苹果。同理，3x + 2x = 5x。但 3 个苹果和 2 根香蕉不能算作一种水果，所以 3x + 2y 不能合并。

For example, 5x + 2y – x can be simplified by combining 5x and -x to get 4x + 2y.

例如，5x + 2y – x 可以合并 5x 与 -x，得到 4x + 2y。

3. Identifying Like Terms | 识别同类项

To identify like terms, check two things: the variables must be the same, and the powers of those variables must be the same.

要识别同类项，需要检查两点：变量必须相同，并且这些变量的次数也必须相同。