Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain several terms that look similar. Learning to collect like terms is the first step toward simplifying algebraic expressions and solving equations.

在代数中，表达式常常包含多个看起来相似的项。学会合并同类项是化简代数表达式和求解方程的第一步。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable part, including the same power. The coefficients can be different.

同类项是指变量部分完全相同（包括相同的幂）的项。它们的系数可以不同。

For example, in the expression 3x + 5x, both terms contain the variable x to the first power, so they are like terms. In contrast, 3x and 3y are not like terms because their variables are different.

例如，在表达式 3x + 5x 中，两项都包含变量 x 的一次幂，因此它们是同类项。相反，3x 和 3y 不是同类项，因为它们的变量不同。

Consider these examples:

请看以下例子：

4a and 9a are like terms.
4a 与 9a 是同类项。
7b² and 3b² are like terms.
7b² 与 3b² 是同类项。
5xy and 2xy are like terms.
5xy 与 2xy 是同类项。
6c and 6d are not like terms.
6c 与 6d 不是同类项。

2. 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 understand. It also prepares the expression for further operations such as substitution, factorisation, or solving equations.

合并同类项可以使表达式更简短、更易于理解。它还为后续的代入、因式分解或解方程等操作做好准备。

For instance, 7x + 2x can be written as 9x. This is simpler and shows the total effect of x at a glance.

例如，7x + 2x 可以写成 9x。这样更简洁，能一眼看出 x 的总效果。

When you collect like terms, you are applying the distributive property in reverse: ax + bx = (a + b)x.

当你合并同类项时，其实是在反向运用分配律：ax + bx = (a + b)x。

ax + bx = (a + b)x

3. Adding and Subtracting Like Terms | 同类项的加减

To add or subtract like terms, simply add or subtract their coefficients. Keep the variable part exactly the same.

加减同类项时，只需对它们的系数进行加减。变量部分保持不变。

Example 1: 5m + 3m = 8m

例 1：5m + 3m = 8m

Example 2: 12p − 4p = 8p

例 2：12p − 4p = 8p

Example 3: 6t + 2t − 3t = 5t

例 3：6t + 2t − 3t = 5t

Notice that the sign before a term belongs to that term. When subtracting 4p, think of it as adding −4p.

注意：项前面的符号属于该项。减去 4p 可以看作加上 −4p。

4. Coefficients and Variables | 系数与变量

The coefficient is the number multiplying the variable. In the term 6x, the coefficient is 6. If a term has no written coefficient, it is understood to be 1.

系数是乘在变量前面的数。在项 6x 中，系数是 6。如果一个项没有写出系数，那么它默认为 1。

For example, y means 1y, and −w means −1w.

例如，y 表示 1y，−w 表示 −1w。

When collecting like terms, always combine the coefficients, not the variables. The variable part remains untouched.

合并同类项时，始终只合并系数，而不要动变量部分。

7x + x = 7x + 1x = 8x

5. Handling Different Variables | 处理不同变量

Terms with different variables cannot be collected. For example, 3x + 4y cannot be simplified further because x and y are different.

变量不同的项不能合并。例如，3x + 4y 不能再化简，因为 x 和 y 不同。

Similarly, terms like 2a + 3b + 4a can only have the a-terms collected: 6a + 3b.

同样，像 2a + 3b + 4a 这样的表达式，只能合并含 a 的项：6a + 3b。

Always look at the variable part carefully. Different letters, or the same letter with different powers, are not like terms.

务必仔细观察变量部分。不同的字母，或者相同字母但幂不同，都不是同类项。

Example: 2x + 3x² − 5x = (2x − 5x) + 3x² = −3x + 3x²

例：2x + 3x² − 5x = (2x − 5x) + 3x² = −3x + 3x²

6. Collecting Terms with Powers | 含幂的同类项

Like terms must have the same variable raised to the same power. Therefore, x² and x are not like terms.

同类项必须具有相同变量且幂也相同。因此，x² 和 x 不是同类项。

For instance, 4k² + 3k² = 7k², but 4k² + 3k cannot be simplified by collecting.

例如，4k² + 3k² = 7k²，但 4k² + 3k 不能通过合并来化简。

When expressions contain powers, group each power separately.

当表达式中含有幂时，要按不同幂分别分组。

5a² + 2a + 3a² − a = (5a² + 3a²) + (2a − a) = 8a² + a

7. Simplifying Expressions with Brackets | 含括号表达式的化简

Sometimes you need to expand brackets before collecting like terms. Use the distributive property to remove the brackets first.

有时需要先展开括号，再合并同类项。使用分配律去掉括号。

Example: 2(3x + 4) + 5x = 6x + 8 + 5x = 11x + 8

例：2(3x + 4) + 5x = 6x + 8 + 5x = 11x + 8

Be careful with negative signs outside brackets: −(2x − 3) = −2x + 3

要小心括号外的负号：−(2x − 3) = −2x + 3

Always expand fully before collecting like terms, so you do not miss any terms.

一定要先完全展开，再合并同类项，以免遗漏任何项。

3(2y + 1) − 2(y − 4) = 6y + 3 − 2y + 8 = 4y + 11

8. Common Mistakes to Avoid | 常见错误

Mistake 1: Combining different variables. For example, 5x + 5y is not 10xy.

错误 1：合并不同变量。例如，5x + 5y 不是 10xy。

Mistake 2: Changing the power when collecting. For example, x² + x² = 2x², not 2x⁴.

错误 2：合并时改变了幂。例如，x² + x² = 2x²，而不是 2x⁴。

Mistake 3: Forgetting the coefficient 1. Remember that x is 1x, and −y is −1y.

错误 3：忘记系数 1。记住 x 就是 1x，−y 就是 −1y。

Mistake 4: Dropping negative signs. Always keep the sign that comes before each term.

错误 4：漏掉负号。始终保留每项前面的符号。

9. Worked Examples | 例题解析

Example A: Simplify 7h + 2k − 3h + 5k

例题 A：化简 7h + 2k − 3h + 5k

Collect the h-terms: 7h − 3h = 4h. Collect the k-terms: 2k + 5k = 7k. So the answer is 4h + 7k.

合并 h 项：7h − 3h = 4h。合并 k 项：2k + 5k = 7k。因此答案为 4h + 7k。

Example B: Simplify 4a + 3b − 2a + b

例题 B：化简 4a + 3b − 2a + b

(4a − 2a) = 2a. (3b + b) = 4b. Answer: 2a + 4b.

(4a − 2a) = 2a。(3b + b) = 4b。答案：2a + 4b。

Example C: Simplify 5x² + 2x − x² + 3x

例题 C：化简 5x² + 2x − x² + 3x

(5x² − x²) = 4x². (2x + 3x) = 5x. Answer: 4x² + 5x.

(5x² − x²) = 4x²。(2x + 3x) = 5x。答案：4x² + 5x。

10. Practice Questions | 练习题

Try these questions yourself, then check your answers.

请先自己尝试以下题目，再核对答案。

Question	题目
1. Simplify 6m + 2m − 3m	1. 化简 6m + 2m − 3m
2. Simplify 4x + 3y + 2x − y	2. 化简 4x + 3y + 2x − y
3. Simplify 3a² + 2a + 5a² − a	3. 化简 3a² + 2a + 5a² − a
4. Simplify 5(2p + 1) − 3p	4. 化简 5(2p + 1) − 3p

Answers: 1) 5m 2) 6x + 2y 3) 8a² + a 4) 7p + 5

答案：1) 5m 2) 6x + 2y 3) 8a² + a 4) 7p + 5

11. Summary | 总结

Collecting like terms is a fundamental algebra skill. Identify terms with the same variable and power, then combine only their coefficients while keeping the variable part unchanged.

合并同类项是一项基础的代数技能。找出具有相同变量和幂的项，然后只合并它们的系数，并保持变量部分不变。

Always expand brackets first, watch out for signs, and never mix different variables or different powers. With practice, this process becomes quick and accurate.

始终先展开括号，注意符号，并且绝不混合不同变量或不同幂。通过练习，这个过程会变得快速而准确。

Mastering this skill will help you solve equations, simplify expressions, and succeed in IGCSE Mathematics.

掌握这一技能将帮助你解方程、化简表达式，并在 IGCSE 数学中取得成功。

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Collecting Like Terms | 合并同类项