Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, expressions are built from terms. Learning to combine like terms is one of the most essential skills for simplifying expressions and solving equations. This article explains what like terms are, how to combine them correctly, and where this skill is used in IGCSE mathematics.

在代数中，表达式由项构成。学会合并同类项是化简表达式和求解方程最重要的基础技能之一。本文将解释什么是同类项、如何正确合并它们，以及在 IGCSE 数学中如何运用这一技能。

1. What Are Like Terms? | 什么是同类项？

A term is a single number, a variable, or a product of numbers and variables. For example, in the term \(5x\), the number 5 is the coefficient and \(x\) is the variable. Terms that have exactly the same variables raised to exactly the same powers are called like terms. Only their coefficients may be different.

项是一个单独的数字、变量或数字与变量的乘积。例如，在项 5x 中，数字 5 是系数，x 是变量。如果几个项的变量完全相同，并且每个变量的指数也完全相同，那么它们就是同类项。只有系数可以不同。

For instance, \(3x\) and \(-2x\) are like terms, because both contain the variable \(x\) to the first power. However, \(3x\) and \(3x^2\) are not like terms, because the powers of \(x\) are different.

例如，3x 和 -2x 是同类项，因为它们都含有变量 x 的一次幂。但是，3x 和 3x² 不是同类项，因为 x 的指数不同。

2. Why Combining Works | 为什么可以合并？

Combining like terms is based on the distributive property of multiplication over addition. This property tells us that:

合并同类项的依据是乘法对加法的分配律。分配律告诉我们：

a x + b x = (a + b) x

Think of \(x\) as “an object”. If you have 3 apples and 5 apples, you have 8 apples in total. In the same way, \(3x + 5x\) means 3 groups of \(x\) plus 5 groups of \(x\), which gives 8 groups of \(x\).

可以把 x 想象成一个“物体”。如果你有 3 个苹果和 5 个苹果，你总共有 8 个苹果。同样地，3x + 5x 表示 3 组 x 加上 5 组 x，结果是 8 组 x。

3. Identifying Like Terms | 识别同类项

To combine like terms, you must first identify them correctly. Like terms must have both the same variables and the same exponents.

要合并同类项，首先必须正确识别它们。同类项必须具有相同的变量和相同的指数。

Same variable and same power: \(2x\) and \(7x\) are like terms.

变量和指数都相同：2x 和 7x 是同类项。
Same variable but different power: \(x\) and \(x^2\) are not like terms.

变量相同但指数不同：x 和 x² 不是同类项。
Different variables: \(3x\) and \(3y\) are not like terms.

变量不同：3x 和 3y 不是同类项。
Constants are like terms: \(5\), \(-8\), and \(1\) are all like terms because they have no variable part.

常数项是同类项：5、-8 和 1 都是同类项，因为它们没有变量部分。

The table below shows examples of how to group like terms in an expression.

下表展示了如何在表达式中对同类项进行分组。

Expression	Like terms	Not like terms
4x + 2x	4x, 2x	—
3x² + x²	3x², x²	—
5x + 5	—	5x, 5
2xy + 3x	—	2xy, 3x

4. Step-by-Step Method | 分步方法

To combine like terms in any expression, follow these steps:

要在任意表达式中合并同类项，请按以下步骤进行：

Write out the expression and identify all like terms.

写出表达式，并找出所有同类项。
Group the like terms together, usually by rearranging the terms.

重新排列项，把同类项放在一起。
Add or subtract the coefficients of each group of like terms.

对每组同类项的系数进行加减运算。
Keep the variable part unchanged.

变量部分保持不变。
Write the simplified expression in descending order of powers when possible.

按指数从高到低的顺序写出化简后的表达式（如可能）。

For example, simplify \(4x + 3 – 2x + 5\). Group the \(x\) terms and the constants:

例如，化简 4x + 3 – 2x + 5。先将 x 项和常数项分组：

4x – 2x + 3 + 5 = 2x + 8

5. Simple Algebraic Examples | 简单代数示例

Let us look at several examples of combining like terms in linear expressions.

我们来看几个在线性表达式中合并同类项的示例。

Example 1: Simplify \(3x + 5x\).

示例 1：化简 3x + 5x。

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

Example 2: Simplify \(7x – 2x + 4\).

示例 2：化简 7x – 2x + 4。

7x – 2x + 4 = 5x + 4

Example 3: Simplify \(2x^2 + 3x – x^2 + x\).

示例 3：化简 2x² + 3x – x² + x。

(2x^2 – x^2) + (3x + x) = x^2 + 4x

Notice that we combine \(x^2\) terms with \(x^2\) terms and \(x\) terms with \(x\) terms separately.

注意，我们把 x² 项与 x² 项合并，x 项与 x 项分别合并。

6. Combining with More Than One Variable | 包含多个变量的合并

When an expression has two or more different variables, we combine terms with exactly the same variable combinations.

当一个表达式包含两个或多个不同变量时，我们只合并变量组合完全相同的项。

Example 4: Simplify \(3x + 2y – x + 4y\).

示例 4：化简 3x + 2y – x + 4y。

(3x – x) + (2y + 4y) = 2x + 6y

Example 5: Simplify \(2xy + 3x + y – xy\).

示例 5：化简 2xy + 3x + y – xy。

(2xy – xy) + 3x + y = xy + 3x + y

In Example 5, \(2xy\) and \(-xy\) are like terms because both contain the same product \(xy\), but \(3x\) and \(y\) are not like terms with \(xy\).

在示例 5 中，2xy 和 -xy 是同类项，因为它们都含有相同的乘积 xy；但 3x 和 y 都不是 xy 的同类项。

7. Removing Parentheses First | 先去括号再合并

Sometimes an expression contains brackets. The brackets must be removed first using the distributive property before combining like terms.

有时表达式中含有括号。必须先用分配律去掉括号，然后再合并同类项。

Example 6: Simplify \(2(x + 3) + 4(x – 1)\).

示例 6：化简 2(x + 3) + 4(x – 1)。

First expand each bracket:

首先展开每个括号：

2(x + 3) = 2x + 6, 4(x – 1) = 4x – 4

Now combine like terms:

然后合并同类项：

2x + 6 + 4x – 4 = 6x + 2

Always be careful with negative signs when expanding brackets.

展开括号时，要特别注意负号。

8. Common Pitfalls to Avoid | 常见错误与避坑

Students often make the following mistakes when combining like terms. Avoid them to improve accuracy.

学生在合并同类项时经常会犯以下错误。避免这些错误能提高准确率。

Combining unlike terms: writing \(3x + 4y = 7xy\) is wrong. The variables must match exactly.

合并非同类项：把 3x + 4y 写成 7xy 是错误的。变量必须完全一致。
Mixing powers: \(x + x^2\) cannot be simplified to \(x^2\) or \(2x^2\).

混淆指数：x + x² 不能化简为 x² 或 2x²。
Sign errors: when subtracting, remember to change the sign of every term inside the bracket. For example, \(3x – (2x – 1) = 3x – 2x + 1 = x + 1\).

符号错误：做减法时，记得改变括号内每一项的符号。例如，3x – (2x – 1) = 3x – 2x + 1 = x + 1。
Forgetting constant terms: constants should also be combined. In \(4x + 3 – 2x + 5\), the constants 3 and 5 combine to 8.

忘记常数项：常数项也应该合并。在 4x + 3 – 2x + 5 中，常数 3 和 5 合并为 8。

9. Using Combining to Solve Equations | 用合并同类项解方程

Combining like terms is often the first step in solving linear equations. It makes the equation simpler and easier to solve.

合并同类项通常是解线性方程的第一步。它能让方程更简单、更容易求解。

Example 7: Solve \(2x + 3 + x = 9\).

示例 7：解方程 2x + 3 + x = 9。

First combine like terms on the left-hand side:

首先合并左边的同类项：

3x + 3 = 9

Then subtract 3 from both sides:

然后两边同时减去 3：

3x = 6

Finally divide by 3:

最后两边同时除以 3：

x = 2

Without combining like terms first, this equation would be more tedious to solve.

如果不先合并同类项，这个方程解起来会更加繁琐。

10. Practice Questions | 练习题目

Try these questions yourself. Simplify each expression fully.

请自己尝试完成以下题目，并化简每一个表达式。

\(5a + 2a\)

5a + 2a
\(4x^2 + 3x – x^2 + 6x\)

4x² + 3x – x² + 6x
\(2m + 3n – m + 5n\)

2m + 3n – m + 5n
\(3(2x + 1) + 2(x – 4)\)

3(2x + 1) + 2(x – 4)
\(7xy – 2x + 3xy + 4x\)

7xy – 2x + 3xy + 4x

Answers: 1. \(7a\) 2. \(3x^2 + 9x\) 3. \(m + 8n\) 4. \(8x – 5\) 5. \(10xy + 2x\)

答案：1. 7a 2. 3x² + 9x 3. m + 8n 4. 8x – 5 5. 10xy + 2x

11. Summary | 总结

Combining like terms is a fundamental algebraic skill that appears throughout IGCSE mathematics. Remember that only terms with the same variable parts and the same exponents can be combined. Group the terms, add or subtract their coefficients, and keep the variable parts unchanged. Always remove brackets first and be watchful of signs.

合并同类项是一项基础代数技能，贯穿于 IGCSE 数学的整个学习过程。请记住：只有变量部分和指数完全相同的项才能合并。将项分组，对其系数进行加减，并保持变量部分不变。始终先去除括号，并注意符号。

With regular practice, you will be able to simplify expressions quickly and accurately, making later equation-solving much easier.

通过经常练习，你将能够快速而准确地化简表达式，让之后的解方程变得更加轻松。

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Combining Like Terms | 合并同类项