Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, expressions often contain multiple terms. Combining like terms is the process of simplifying an algebraic expression by adding or subtracting terms that have exactly the same variable parts. This is one of the most fundamental skills you will need for IGCSE Mathematics, as it appears in everything from linear equations to quadratic expressions and beyond.

在代数中，表达式通常包含多个项。合并同类项是通过加减具有完全相同变量部分的项来简化代数表达式的过程。这是 IGCSE 数学中最基础的技能之一，从线性方程到二次表达式等几乎所有内容都会用到它。

1. What Are Like Terms? | 什么是同类项

Like terms are terms whose variables (and their exponents) are exactly the same. The coefficients of the terms can be different, but the variable part must match completely. Only like terms can be added or subtracted together.

同类项是指变量（及其指数）完全相同的项。各项的系数可以不同，但变量部分必须完全匹配。只有同类项才能进行加减运算。

Examples: 4x and 7x are like terms; 3y and −2y are like terms; 5x² and 2x² are like terms. However, 4x and 4y are not like terms, and 3x² and 3x are not like terms.

示例：4x 和 7x 是同类项；3y 和 −2y 是同类项；5x² 和 2x² 是同类项。但是，4x 和 4y 不是同类项，3x² 和 3x 也不是同类项。

Like Terms	Not Like Terms
2a, 5a, −7a	2a, 2b, 2c
x², 3x², −½x²	x², x, 5
4pq, −pq, 2pq	4pq, 4p, q

2. Why Combine Like Terms? | 为什么合并同类项

Combining like terms simplifies an expression, making it easier to evaluate, substitute values into, or solve equations containing the expression. A simplified expression is shorter, produces fewer possible errors, and is more readily comparable to other expressions.

合并同类项可以简化表达式，使其更易于求值、代入数值或解含有该表达式的方程。简化后的表达式更简短，能减少出错的概率，也更容易与其他表达式进行比较。

For example, the expression 6x + 3x − 2x + 4x might look complicated at first, but after combining all the like terms we simply get 11x.

例如，表达式 6x + 3x − 2x + 4x 乍看起来可能很复杂，但合并所有同类项后，我们直接得到 11x。

3. The Coefficient and Variable Structure | 系数与变量结构

Every term in an algebraic expression is composed of two parts: a coefficient (the numerical factor) and a variable part (letters and their powers). When combining like terms, we only add or subtract the coefficients while keeping the variable part unchanged.

代数表达式中的每一项都由两部分组成：系数（数字因数）和变量部分（字母及其幂次）。合并同类项时，我们只对系数进行加减，保持不变的是变量部分。

ax ⁿ + bx ⁿ = (a + b)x ⁿ

Here, a and b are coefficients, and xⁿ is the common variable part. The variable part xⁿ must be identical for the two terms to be combinable.

其中 a 和 b 是系数，xⁿ 是共同的变量部分。只有当两项的变量部分 xⁿ 完全相同时，它们才可以合并。

4. Basic Addition and Subtraction | 基础加减运算

When adding or subtracting like terms, simply combine their coefficients. The sign in front of each term must be carried along with it.

在加减同类项时，只需合并它们的系数。每项前面的符号必须随该项一起移动。

Example 1: Simplify 3x + 5x.

示例 1：化简 3x + 5x。

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

Example 2: Simplify 9y − 4y.

示例 2：化简 9y − 4y。

9y − 4y = (9 − 4)y = 5y

Example 3: Simplify 2a + 3a − 5a.

示例 3：化简 2a + 3a − 5a。

2a + 3a − 5a = (2 + 3 − 5)a = 0a = 0

Notice that the result 0a is simply written as 0.

请注意，结果 0a 通常直接写成 0。

5. Combining Terms with Different Variables | 合并含不同变量的项

Terms that contain different variables are not like terms and cannot be combined. For example, 4x and 4y must remain separate terms in a simplified expression. When an expression contains several groups of like terms, combine each group independently.

含有不同变量的项不是同类项，不能合并。例如，4x 和 4y 在简化表达式中必须保持为独立的项。当表达式中包含多组同类项时，应分别合并每一组。

Example: Simplify 5x + 2y − 3x + 4y.

示例：化简 5x + 2y − 3x + 4y。

First, group the x terms and the y terms separately.

首先，分别将 x 项和 y 项分组。

5x − 3x + 2y + 4y = 2x + 6y

The final answer is 2x + 6y. No further simplification is possible because 2x and 6y are not like terms.

最终答案为 2x + 6y。由于 2x 和 6y 不是同类项，因此无法进一步化简。

6. Combining Terms with Exponents | 合并含指数的项

Terms with the same variable but different powers are not like terms. For instance, x² and x³ cannot be combined, nor can x² and x. Only terms with the exact same variable and the exact same exponent can be added or subtracted.

变量相同但幂次不同的项不是同类项。例如，x² 和 x³ 不能合并，x² 和 x 也不能合并。只有变量相同且指数完全相同的项才可以进行加减。

Example 1: Simplify 4x² + 3x².

示例 1：化简 4x² + 3x²。

4x² + 3x² = 7x²

Example 2: Simplify x³ + 2x² − 3x³ + x².

示例 2：化简 x³ + 2x² − 3x³ + x²。

Group the x³ terms and the x² terms separately.

分别将 x³ 项和 x² 项分组。

x³ − 3x³ + 2x² + x² = −2x³ + 3x²

Remember: −2x³ and 3x² cannot be combined because their exponents differ.

请记住：−2x³ 和 3x² 不能合并，因为它们的指数不同。

7. Combining Terms with Multiple Variables | 合并含多个变量的项

When terms have more than one variable, the entire variable string must match. For example, 3ab and 5ab are like terms, but 3ab and 3a²b are not like terms.

当项含有多个变量时，整个变量串必须完全匹配。例如，3ab 和 5ab 是同类项，但 3ab 和 3a²b 不是同类项。

Example: Simplify 2ab + 3a²b − ab + 4a²b.

示例：化简 2ab + 3a²b − ab + 4a²b。

Group terms with identical variable parts.

将具有相同变量部分的项分组。

2ab − ab + 3a²b + 4a²b = ab + 7a²b

Note that ab and a²b cannot be combined because the exponent of a differs.

请注意，ab 和 a²b 不能合并，因为 a 的指数不同。

8. Combining Like Terms after Expanding Brackets | 去括号后合并同类项

Often, an expression contains brackets that must be expanded first. Once the brackets are removed, you can then combine like terms to simplify fully.

通常，表达式中包含括号，必须先进行展开。当括号去掉后，再合并同类项以完成化简。

Example: Simplify 2(x + 3) + 3(x + 1).

示例：化简 2(x + 3) + 3(x + 1)。

Step 1: Expand both brackets.

步骤 1：展开两个括号。

2x + 6 + 3x + 3

Step 2: Combine the x terms and the constant terms.

步骤 2：合并 x 项和常数项。

2x + 3x + 6 + 3 = 5x + 9

The final simplified expression is 5x + 9.

最终简化表达式为 5x + 9。

9. Common Mistakes and How to Avoid Them | 常见错误及如何避免

The following table shows common mistakes students make when combining like terms, along with the correct approach.

下表列出了学生在合并同类项时常犯的错误以及正确的做法。

Common Mistake	Correct Calculation
x + x² = 2x³	x + x² cannot be simplified further
3x + 2y = 5xy	3x + 2y stays as 3x + 2y
5a − 3a² = 2a	5a − 3a² stays unchanged
2(x + 4) = 2x + 4	2(x + 4) = 2x + 8

To avoid these errors, always check three things: the variable is the same, the exponent of each variable is the same, and you have correctly applied the distributive property when expanding brackets.

要避免这些错误，请务必检查三点：变量相同、每个变量的指数相同、并且在展开括号时正确应用了分配律。

10. Practice Problems | 练习

Try these problems on your own, then check your answers below.

请先独立尝试以下题目，然后对照下面的答案。

1. Simplify 7x + 2x − 5x.

1. 化简 7x + 2x − 5x。

2. Simplify 4a + 3b − 2a + 6b.

2. 化简 4a + 3b − 2a + 6b。

3. Simplify 3x² + 2x − x² + 5x.

3. 化简 3x² + 2x − x² + 5x。

4. Simplify 5(2y + 1) − 2(3y − 4).

4. 化简 5(2y + 1) − 2(3y − 4)。

5. Simplify 6mn − 2m²n + 3mn + m²n.

5. 化简 6mn − 2m²n + 3mn + m²n。

Answers:

答案：

Problem	Answer
1	4x
2	2a + 9b
3	2x² + 7x
4	4y + 13
5	9mn − m²n

11. Exam Tips | 考试技巧

In IGCSE examinations, always show your working clearly. When simplifying expressions, write each step on a new line rather than trying to do everything mentally. Check the signs of each term carefully, especially when subtracting a bracket. Finally, always check whether your final answer can be simplified further.

在 IGCSE 考试中，务必清晰地展示你的解题步骤。化简表达式时，每一步都写在新的一行，不要试图心算完成所有步骤。仔细检查每项的符号，尤其是在减去括号时。最后，始终检查你的最终答案是否还能进一步化简。

5x + 3y − 2x + y = 3x + 4y

A neat, step-by-step presentation helps you earn method marks even if you make a small arithmetic slip at the end.

整洁、逐步的书写方式可以帮助你在最后出现小计算失误时仍然获得步骤分。

12. Summary | 总结

Combining like terms is a core skill in algebra. Like terms have identical variable parts, including matching exponents. When combining, add or subtract only the coefficients while keeping the variable part unchanged. Terms with different variables or different exponents cannot be combined. Expanding brackets before combining is often necessary, and careful sign handling is essential.

合并同类项是代数中的核心技能。同类项具有完全相同的变量部分，包括匹配的指数。合并时，只对系数进行加减，保持变量部分不变。变量不同或指数不同的项不能合并。在合并前通常需要先展开括号，并且仔细处理符号至关重要。

Mastering this skill will make solving equations, simplifying expressions, and working with functions much easier in both Paper 1 and Paper 2 of the IGCSE Mathematics exam.

掌握这项技能将使你在 IGCSE 数学卷 1 和卷 2 中解方程、化简表达式以及处理函数时都更加得心应手。

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Combining Like Terms | 合并同类项