Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Collecting like terms is one of the most important algebra skills in the IGCSE Mathematics course. It is the process of simplifying expressions by adding or subtracting terms that have exactly the same variable structure.

合并同类项是 IGCSE 数学课程中最最重要的代数技能之一。它是指把代数表达式中变量结构完全相同的项通过加法或减法合并起来，从而化简表达式的过程。

1. What Is a Term? | 什么是“项”？

In algebra, a term is a single mathematical expression separated by the + or − sign in a longer expression. A term can be a number, a variable, or a product of numbers and variables.

在代数中，“项”是在一个较长的表达式中被加号或减号隔开的单个数学表达式。一个项可以是一个数、一个字母，或者是数与字母的乘积。

For example, the expression 2x + 3 − 5y² has three terms: 2x, 3 and −5y².

例如，表达式 2x + 3 − 5y² 中共有三项：2x、3 和 −5y²。

2. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part. This means the variables are the same, and each variable is raised to the same power. Only the numerical coefficients may be different.

同类项是指变量部分完全相同的项。也就是说，所含的字母相同，并且相同字母的指数也相同。只有数字系数可以不同。

For example, 3x and −7x are like terms because both contain the single variable x. Similarly, 2y² and ½y² are like terms because both contain y².

例如，3x 和 −7x 是同类项，因为二者都含有单个变量 x。同样，2y² 和 ½y² 也是同类项，因为二者都含有 y²。

Unlike terms have a different variable or a different power. For example, 3x and 3x² are not like terms because the power of x is different.

非同类项的变量不同或次数不同。例如，3x 和 3x² 不是同类项，因为 x 的指数不同。

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

Collecting like terms makes an expression shorter and easier to work with. It is also needed before solving equations and for writing final answers clearly.

合并同类项可以使表达式更简短，更便于处理。同时，在解方程之前以及写出简洁的最终答案时，都需要先合并同类项。

4a + 2 + 3a − 1 = 7a + 1

The original expression has four terms, but after collecting like terms it has only two. This is much easier to substitute into or to graph.

原表达式中共有四项，而合并同类项之后只剩下两项。这样的形式无论代入数值还是作图，都更加简便。

4. The Rule: The Distributive Property | 合并规则：乘法分配律

The reason collecting like terms works is the distributive property. Because both terms share the same variable, we can factor the variable out and add the coefficients.

合并同类项能够成立的原因是乘法分配律。因为各项含有相同的变量，我们可以把变量提出来，只对系数进行加减。

3x + 5x = (3 + 5)x = 8x

Similarly, subtraction follows the same pattern:

减法也同样适用这一规律：

7a − 2a = (7 − 2)a = 5a

5. Combining Positive and Negative Terms | 正项与负项的合并

When a term is subtracted, it is better to view the sign as part of that term. For example, in the expression 7x − 4x, the second term is −4x.

当一项被减去时，最好把符号看成那一项的一部分。例如，在表达式 7x − 4x 中，第二项是 −4x。

Work out the coefficients with their signs:

计算系数时要把符号一起考虑：

7x − 4x = (7 − 4)x = 3x

For negative results, keep the sign in front:

如果结果为正，保留正号；如果结果为负，保留负号：

−3a + 2a = (−3 + 2)a = −a

6. Powers Must Match | 次数必须一致

A very common error is to combine x with x². They are not like terms because the variables have different powers. You cannot add x and x² to obtain x³ or any other single term.

一个非常常见的错误是把 x 与 x² 合并。它们不是同类项，因为变量的次数不同。你不能把 x 和 x² 相加得到 x³，也不能得到一个单项式。

However, x² and 2x² are like terms, so they can be collected:

但是，x² 和 2x² 是同类项，因此可以合并：

x² + 2x² = 3x²

Similarly, 4x³ and 5x³ are like terms, but 4x³ and 5x² are not.

同样，4x³ 和 5x³ 是同类项，但 4x³ 和 5x² 不是同类项。

7. Collecting Terms with Several Variables | 含多个变量的同类项

When terms contain more than one variable, every variable and its power must match. For example, 3mn and −5mn are like terms because both contain m and n.

当项中含有多个变量时，每一个变量及其指数都必须相同。例如，3mn 和 −5mn 是同类项，因为它们都含有 m 和 n。

The order of multiplication does not matter, so xy and yx are the same type of term.

乘法顺序不影响结果，因此 xy 和 yx 是同一类项。

4ab + 3a − 2ab + b = 2ab + 3a + b

Note that 3a and b cannot be collected with 2ab, because ab is different from a and from b.

注意，3a 和 b 都不能与 2ab 合并，因为 ab 与单独的 a、b 都不相同。

8. Collecting Like Terms in Bracket Expressions | 括号表达式中的合并

If an expression contains brackets, expand the brackets first, then collect like terms. This is a frequent IGCSE exam question format.

如果表达式含有括号，要先展开括号，再合并同类项。这是 IGCSE 考试中经常出现的题型。

3(x + 2) + 2(x − 1)

= 3x + 6 + 2x − 2

= 5x + 4

In the final step, 3x and 2x are like terms, while 6 and −2 are constant like terms.

在最后一步，3x 和 2x 是同类项，而 6 和 −2 是常数同类项。

9. Common Mistakes | 常见错误

Here are the most common mistakes students make when collecting like terms.

以下是学生在合并同类项时最常见的错误。

Adding different powers: x + x² cannot be simplified further. It is not x³.

把不同次数的项相加：x + x² 不能再化简，它不是 x³。
Forgetting the negative sign: 6x − 4x is 2x, not 6x − 4 giving 2 without x.

遗漏负号：6x − 4x 等于 2x，并不是 6 − 4 得到 2 然后丢掉 x。
Writing x + x = x²: x + x is 2x. The expression x × x is x².

把 x + x 写成 x²：x + x 等于 2x。只有 x × x 才等于 x²。
Mixing different variables: 3a + 3b cannot be simplified into one term.

混淆不同变量：3a + 3b 不能简化成单项式。

10. Worked Examples | 典型例题

Study these worked examples carefully before trying your own questions.

请先认真研究以下典型例题，然后再自己尝试解题。

Expression 表达式	Simplified 化简结果
2a + 5b − a + 3b	a + 8b
x² + 3x − 2x² + 4x	−x² + 7x
5 + 2p − 3 + 4p	6p + 2
3(x + 1) + 2(x − 4)	5x − 5

In the first example, collect a-terms and b-terms separately. In the fourth example, expand the brackets before collecting.

在第一个例题中，分别合并含 a 的项和含 b 的项。在第四个例题中，要先展开括号，再合并同类项。

11. Exam-Style Practice | 考试风格练习

Try these questions in about five minutes. Simplify each expression fully.

请在五分钟内完成以下题目，并将每个表达式完整化简。

Question 题目	Answer 答案
4x + 3x − 2x	5x
7y + 2 − 3y + 5	4y + 7
2x² + 3x − x² + x	x² + 4x
4(m + 2) − 2(m − 3)	2m + 14

If you made a mistake, check whether you matched the variables and powers correctly before looking at the answer.

如果做错了，先检查自己在合并之前是否正确判断了变量和次数是否相同，然后再看答案。

12. Exam Tips | 应试提示

Always write your final answer in descending powers if possible. For example, write x² + 4x rather than 4x + x².

在写最终答案时，尽量按未知数次数从高到低排列。例如，写 x² + 4x，而不是 4x + x²。

When a variable appears alone, remember it has coefficient 1:

如果变量单独出现，要记住它的系数是 1：

x + 3x = 4x

Check each term as you move it: bring its sign with it, and do not change the power of the variable.

每移动一项都要检查：符号要跟着项一起移动，并且不能改变变量的指数。

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Collecting Like Terms | 合并同类项