Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

At IGCSE level, algebra questions often ask you to simplify expressions by collecting like terms. This skill is used in solving equations, expanding brackets and rearranging formulas. Mastering it early makes the whole algebra topic much easier.

在 IGCSE 阶段，代数题常要求通过合并同类项来化简表达式。这一技能在解方程、去括号和公式变形中都会用到。尽早掌握它会让整个代数专题变得简单很多。

1. What Are Like Terms? | 什么是同类项

A term is a number, a variable, or a product of numbers and variables. Like terms are terms that have exactly the same variable part raised to the same powers. The numbers in front, called coefficients, can be different.

项是一个数、一个变量，或数与变量的乘积。同类项是指变量部分完全相同且次数相同的项。前面的数字叫系数，系数可以不同。

For example, 5x and −3x are like terms because both contain the variable x to the power 1. 2x² and 7x² are also like terms because both contain x².

例如，5x 和 −3x 是同类项，因为它们都含有一次方的变量 x。2x² 和 7x² 也是同类项，因为它们都含有 x²。

2. Why Collect Like Terms? | 为什么要合并同类项

Collecting like terms makes an expression shorter and easier to use. For example, 3x + 2y + 5x − y can be simplified to 8x + y. The simplified form has fewer terms and is much cleaner.

合并同类项能让表达式更短、更易使用。例如，3x + 2y + 5x − y 可以化简为 8x + y。化简后的形式项数更少，也更整洁。

IGCSE exam papers often give an expression such as 4a + 3b − 2a + 6b and ask you to simplify it fully. You should combine each group of like terms separately and present the final answer clearly.

IGCSE 试卷中常给出如 4a + 3b − 2a + 6b 的表达式，要求完整化简。你应当把每一组同类项分别合并，并清晰写出最终答案。

3. Identifying Like Terms | 识别同类项

Check two things before combining terms: the variables must match, and the powers of each variable must match. Numbers without a variable are called constants, and all constants are like terms with each other.

在合并之前要检查两点：变量必须相同，且每个变量的次数必须相同。不含变量的数叫常数，所有常数之间都是同类项。

Like: 6x, −2x, 9x
Like: 4xy, 11xy, −xy
Not like: 3x and 3x²

同类项：6x、−2x、9x；同类项：4xy、11xy、−xy；不是同类项：3x 和 3x²

4. The Golden Rule: Same Variables, Same Powers | 黄金法则：相同变量且相同次数

The golden rule is simple: only terms with exactly the same variable part can be added or subtracted. x and x² are not the same, because the powers are different.

黄金法则很简单：只有变量部分完全相同的项才能相加或相减。x 和 x² 不相同，因为次数不同。

Similarly, xy and x²y are not like terms. However, ab and ba are like terms because multiplication is commutative, so the order of variables does not matter.

同理，xy 和 x²y 不是同类项。但 ab 和 ba

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