Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, simplifying expressions is one of the most important skills you need. Collecting like terms is the process of grouping and adding terms that have exactly the same variable parts. This makes expressions shorter and easier to work with.

在代数学中，化简表达式是最重要的基本技能之一。合并同类项就是将变量部分完全相同的项分组相加，使表达式更加简短、更易处理。

1. What are Algebraic Expressions? | 什么是代数表达式？

An algebraic expression is a combination of numbers, letters, and operation signs. For example, 3x + 2y – 5 is an expression. Each part separated by a plus or minus sign is called a term.

代数表达式是由数字、字母和运算符号组成的组合。例如，3x + 2y – 5 就是一个表达式。每个用加号或减号隔开的部分称为一个项。

In the expression 3x + 2y – 5, the terms are: 3x, 2y, and -5. The number in front of a variable is called the coefficient.

在表达式 3x + 2y – 5 中，各项分别为：3x、2y 和 -5。变量前面的数字称为系数。

2. What are Like Terms? | 什么是同类项？

Like terms are terms whose variable parts are identical. This means the same letter or letters raised to exactly the same powers. For example, 2x and 5x are like terms because both contain the variable x to the power 1.

同类项是指变量部分完全相同的项，即相同的字母且幂指数完全相同。例如，2x 和 5x 是同类项，因为两者都含有变量 x，且指数为1。

However, 2x and 2y are not like terms because they have different variables. Also, 2x and 2x² are not like terms because the powers of x are different.

但是，2x 和 2y 不是同类项，因为它们的变量不同；同样，2x 和 2x² 也不是同类项，因为 x 的幂指数不同。

3. The Rule of Combining Like Terms | 合并同类项的规则

To combine like terms, simply add or subtract their coefficients while keeping the variable part unchanged. For example:

合并同类项的规则是：只对系数进行加或减，变量部分保持不变。例如：

2x + 3x = (2 + 3)x = 5x

Similarly, 7a – 2a = 5a. You never change the variable part when combining like terms.

同理，7a – 2a = 5a。合并同类项时，你永远不能改变变量部分。

4. Dealing with Positive and Negative Signs | 处理正负号

When combining terms, pay careful attention to the sign in front of each term. The sign belongs to the term that follows it.

合并项时，要特别注意每一项前面的符号。这个符号属于它后面的那个项。

For example, in 5x – 3x + 2x, you have coefficients +5, -3, and +2. Combine them: 5 – 3 + 2 = 4, so the result is 4x.

例如，在 5x – 3x + 2x 中，系数分别为 +5、-3 和 +2。将它们合并：5 – 3 + 2 = 4，所以结果是 4x。

If all coefficients are negative, the result is negative. For instance, -4b – 6b = -10b.

如果所有系数都是负的，结果就是负的。例如，-4b – 6b = -10b。

5. Terms with Different Powers | 不同次幂的项

You can only combine terms with the exact same power. For instance, x² and x cannot be combined into a single term.

你只能合并幂完全相同的项。例如，x² 和 x 不能合并成一个项。

Consider the expression 3x² + 2x + 5x² – x. First collect the x² terms: 3x² + 5x² = 8x². Then collect the x terms: 2x – x = x. The simplified result is 8x² + x.

考虑表达式 3x² + 2x + 5x² – x。先合并 x² 项：3x² + 5x² = 8x²；再合并 x 项：2x – x = x。化简结果为 8x² + x。

Notice that the constant term 4 is also itself. Constants are like terms with other constants. For example, 4 + 7 = 11.

注意，常数项 4 本身也是同类项。常数与其他常数是同类项。例如，4 + 7 = 11。

6. Combining More Than Two Groups | 合并多组同类项

An expression may contain several groups of like terms. A good strategy is to identify each variable and power separately, then combine them in order.

一个表达式可能包含多组同类项。好的策略是分别确定每个变量和幂，然后按顺序合并。

For example, simplify 4m + 3n – 2m + 6n.

例如，化简 4m + 3n – 2m + 6n。

Combine the m terms: 4m – 2m = 2m.

合并 m 项：4m – 2m = 2m。

Combine the n terms: 3n + 6n = 9n.

合并 n 项：3n + 6n = 9n。

Final answer: 2m + 9n.

最终答案：2m + 9n。

7. Expanding Brackets Before Combining | 先展开括号再合并

Sometimes an expression contains brackets. To collect like terms, you must first expand the brackets using the distributive law.

有时表达式中含有括号。要合并同类项，必须先用乘法分配律展开括号。

For example, simplify 3(2x + 1) + 2(x – 4).

例如，化简 3(2x + 1) + 2(x – 4)。

First expand: 3 × 2x + 3 × 1 + 2 × x + 2 × (-4) = 6x + 3 + 2x – 8.

先展开：3 × 2x + 3 × 1 + 2 × x + 2 × (-4) = 6x + 3 + 2x – 8。

Now combine like terms: 6x + 2x = 8x and 3 – 8 = -5. So the simplified expression is 8x – 5.

现在合并同类项：6x + 2x = 8x，3 – 8 = -5。因此化简后的表达式为 8x – 5。

8. Common Mistakes to Avoid | 避免常见错误

Here are some frequent errors students make when collecting like terms:

以下是学生们在合并同类项时常犯的错误：

Mistake 1: Combining 3x and 3. These are not like terms because one has a variable and the other does not.

错误1：将 3x 与 3 合并。它们不是同类项，因为一个含有变量，另一个不含变量。
Mistake 2: Combining 2x and 2x². Remember the powers must match exactly.

错误2：将 2x 与 2x² 合并。记住幂必须完全一致。
Mistake 3: Losing the negative sign. Always carry the sign with the coefficient.

错误3：丢失负号。始终带着该系数前面的正负号。
Mistake 4: Writing 2x + 3y as 5xy. Different variables cannot be combined into a product.

错误4：把 2x + 3y 写成 5xy。不同的变量不能合并成一个乘积。

9. Worked Examples | 例题精讲

Let us work through two typical IGCSE-style questions step by step.

我们通过两道典型的 IGCSE 风格的题目，逐步进行解答。

Example A: Simplify 6p + 2q – 3p + q.

例题A：化简 6p + 2q – 3p + q。

Combine the p terms: 6p – 3p = 3p.

合并 p 项：6p – 3p = 3p。

Combine the q terms: 2q + q = 3q.

合并 q 项：2q + q = 3q。

Answer: 3p + 3q.

答案：3p + 3q。

Example B: Simplify 4(2x – 3) – 2(3x – 1).

例题B：化简 4(2x – 3) – 2(3x – 1)。

Expand: 8x – 12 – 6x + 2.

展开：8x – 12 – 6x + 2。

Combine x terms: 8x – 6x = 2x.

合并 x 项：8x – 6x = 2x。

Combine constants: -12 + 2 = -10.

合并常数项：-12 + 2 = -10。

Answer: 2x – 10.

答案：2x – 10。

10. Practice Questions | 练习

Try these on your own. Simplify each expression fully.

请尝试独立完成以下题目，并将每个表达式化到最简。

Question \| 题目	Answer \| 答案
1. 5x + 7x	12x
2. 9a – 4a + a	6a
3. 3m + 2n – m + 5n	2m + 7n
4. x² + 2x + 3x² – x	4x² + x
5. 2(3x + 4) + 5(x – 2)	11x – 2

11. Summary | 总结

Collecting like terms is a fundamental skill in algebra. Always check that the variable parts are identical before combining, keep the signs attached to each term, and remember that different powers cannot be merged.

合并同类项是代数中的基础技能。合并前务必检查变量部分是否完全相同，注意保留每一项前的符号，并牢记不同幂次不能合并。

With practice, you will be able to simplify expressions quickly and accurately, which will help you in many other topics such as solving equations and expanding brackets.

通过练习，你将能快速准确地化简表达式，这将在解方程、展开括号等其他许多主题中为你提供巨大帮助。

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Collecting Like Terms | 合并同类项