Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, simplifying expressions is one of the first essential skills you will learn. Combining like terms allows you to make expressions shorter, clearer, and easier to work with. Whether you are solving equations, factorising, or rearranging formulas, mastering this technique is a must for IGCSE Mathematics.

在代数学习中，化简表达式是你最早需要掌握的核心技能之一。合并同类项能让表达式更简洁、更清晰，也更容易处理。无论是解方程、因式分解，还是变形公式，掌握这一技巧都是IGCSE数学的必备能力。

1. What Are Like Terms? | 什么是同类项？

An algebraic expression contains terms separated by plus (+) or minus (−) signs. Each term may be a number, a variable, or a product of numbers and variables. Like terms are terms that have exactly the same variable parts, with the same powers. For example, 3x and 5x are like terms because both contain the variable x raised to the same power.

代数表达式由加号（+）或减号（−）连接的若干项组成。每一项可以是一个数、一个字母，或数字与字母的乘积。同类项是指所含变量部分完全相同的项，且相同字母的幂指数也必须相同。例如，3x 和 5x 是同类项，因为它们都含有变量 x，且 x 的指数相同。

Examples of like terms:

同类项示例：

4x and −2x are like terms (variable x, power 1).
4x 和 −2x 是同类项（变量 x，指数为1）。
7y² and 3y² are like terms (variable y, power 2).
7y² 和 3y² 是同类项（变量 y，指数为2）。
5xy and 2xy are like terms (variables x and y, each to the power 1).
5xy 和 2xy 是同类项（变量 x 和 y，且指数都为1）。

Non-examples: 3x and 3x² are not like terms, because the powers of x are different. Similarly, 4x and 4xy are not like terms because the variable parts are different.

反例：3x 和 3x² 不是同类项，因为 x 的指数不同。同样，4x 和 4xy 也不是同类项，因为变量部分不同。

2. Why Do We Combine Like Terms? | 我们为什么要合并同类项？

Combining like terms simplifies an expression into a more compact form. A simplified expression is easier to evaluate, substitute into, or solve. It also helps you see the structure of a problem, such as finding the coefficient of a variable quickly.

合并同类项可以把表达式化为更紧凑的形式。化简后的表达式更容易代入数值、求值或解方程。它还能帮助你看清问题的结构，比如快速找到某个变量的系数。

For example, the expression 5x + 3x can be written as 8x. This is simpler and saves time in later steps. In every algebra problem, simplification is the first step toward finding the answer.

例如，表达式 5x + 3x 可以写成 8x。这样更简洁，也能为后续步骤节省时间。在任何代数问题中，化简都是走向答案的第一步。

3. The Distributive Property and Combining | 分配律与合并

Sometimes an expression contains brackets before we can combine like terms. The distributive property states that a(b + c) = ab + ac. We use this to expand brackets, then we combine like terms if possible.

有时表达式中含有括号，我们需要先去掉括号，再合并同类项。分配律指出：a(b + c) = ab + ac。我们用这个法则展开括号，然后再看是否能合并同类项。

a(b + c) = ab + ac

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

示例：化简 2(x + 3) + x。

First expand: 2(x + 3) = 2x + 6. Then add x: 2x + 6 + x = (2 + 1)x + 6 = 3x + 6.

先展开：2(x + 3) = 2x + 6。再加上 x：2x + 6 + x = (2 + 1)x + 6 = 3x + 6。

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

To combine like terms, add or subtract their coefficients while keeping the variable part unchanged. The coefficient is the number in front of the variable. For example, in the term 7y, the coefficient is 7.

合并同类项时，只需对系数进行加减，变量部分保持不变。系数是变量前面的数。例如，在 7y 中，系数是 7。

ax + bx = (a + b)x

Example: 3x + 5x = (3 + 5)x = 8x.

示例：3x + 5x = (3 + 5)x = 8x。

Example with subtraction: 9y − 4y = (9 − 4)y = 5y.

减法示例：9y − 4y = (9 − 4)y = 5y。

Always remember: you cannot add or subtract terms that are not like terms. For example, 2x + 3y cannot be simplified because x and y are different variables.

切记：不能加减不是同类项的项。例如，2x + 3y 无法化简，因为 x 和 y 是不同的变量。

5. Combining Terms with Different Signs | 合并带不同符号的项

Terms can have positive or negative signs. When combining them, treat the sign as part of the coefficient. For example, −5x means the coefficient is −5.

项可以带有正号或负号。合并时，要把符号视为系数的一部分。例如，−5x 表示系数是 −5。

Example: Simplify 7x − 3x + 2x.

示例：化简 7x − 3x + 2x。

Combine the coefficients: 7 − 3 + 2 = 6. So the expression becomes 6x.

合并系数：7 − 3 + 2 = 6。因此表达式变为 6x。

Here is a longer example: −4a + 3a − a = (−4 + 3 − 1)a = −2a. Notice that an invisible coefficient of 1 before a is treated as −1 when preceded by a minus sign.

再看一个稍长的例子：−4a + 3a − a = (−4 + 3 − 1)a = −2a。注意在 a 前面看不见的系数1，若前面有负号，则按 −1 处理。

6. Simplifying Expressions with Multiple Variables | 化简含多个变量的表达式

When an expression contains more than one variable, group the like terms for each variable separately. Then combine each group.

当表达式包含多个变量时，需要按变量分别分组，然后合并每一组。

Example: Simplify 3a + 2b − a + 4b.

示例：化简 3a + 2b − a + 4b。

Group the a terms: 3a − a = 2a. Group the b terms: 2b + 4b = 6b. So the simplified expression is 2a + 6b.

合并 a 的项：3a − a = 2a。合并 b 的项：2b + 4b = 6b。因此化简结果为 2a + 6b。

Be careful with variables that have different powers. For instance, x and x² are not like terms, so 2x + 3x² cannot be combined. They remain as separate terms.

注意不同幂的变量不能混为一谈。例如，x 和 x² 不是同类项，所以 2x + 3x² 不能合并，它们需要作为不同项保留。

7. Combining Like Terms in Polynomials | 多项式中的合并

A polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, and non-negative integer powers. Simplifying a polynomial means writing it in standard form, usually from the highest power to the lowest.

多项式是由变量和系数组成的表达式，只涉及加法、减法和非负整数次幂。化简多项式通常要写成标准形式，一般按从最高次到最低次的顺序排列。

Example: Simplify 4x² + 3x − 2x² + x.

示例：化简 4x² + 3x − 2x² + x。

Combine x² terms: 4x² − 2x² = 2x². Combine x terms: 3x + x = 4x. So the result is 2x² + 4x.

合并 x² 项：4x² − 2x² = 2x²。合并 x 项：3x + x = 4x。因此结果为 2x² + 4x。

If a polynomial has terms like 5, −3, and 2, these are constant terms and are all like terms. Combine them: 5 − 3 + 2 = 4.

如果多项式中有 5、−3、2 这样的常数项，它们彼此都是同类项。合并后：5 − 3 + 2 = 4。

8. Common Mistakes to Avoid | 避免常见错误

Many students make similar mistakes when combining like terms. Learn these pitfalls so you can avoid them.

很多学生在合并同类项时都会犯一些相似的错误。了解这些陷阱，你就能避开它们。

Mistake 1: Combining terms with different powers, such as 2x + 3x² = 5x³ (wrong). Correct: 2x + 3x² cannot be simplified.
错误1：合并不同幂的项，例如 2x + 3x² = 5x³（错误）。正确：2x + 3x² 无法化简。
Mistake 2: Forgetting the invisible coefficient 1. For example, x + 2x = 3x, not just 2x.
错误2：忘记隐含的系数1。例如 x + 2x = 3x，而不是 2x。
Mistake 3: Ignoring signs. For example, 5x − 3x + 2x is not 0x; it equals 4x.
错误3：忽略符号。例如 5x − 3x + 2x 不是 0x，而等于 4x。
Mistake 4: Combining 3xy and 4yx? Actually they are like terms, but 3xy and 4x² are not.
错误4：把 3xy 和 4yx 误认为不是同类项（它们其实是同类项），但 3xy 和 4x² 不是同类项。

9. Worked Examples | 实例演练

Let us go through two detailed examples step by step.

下面通过两个详细的例子逐步演示。

Example A: Simplify 2(3x − 1) + 4x.

示例A：化简 2(3x − 1) + 4x。

Step 1: Expand the brackets.

步骤1：展开括号。

2(3x − 1) = 6x − 2.

2(3x − 1) = 6x − 2。

Step 2: Rewrite the expression: 6x − 2 + 4x.

步骤2：重写表达式：6x − 2 + 4x。

Step 3: Combine the x terms: 6x + 4x = 10x. The constant is −2. Final answer: 10x − 2.

步骤3：合并 x 项：6x + 4x = 10x。常数项是 −2。最终答案：10x − 2。

Example B: Simplify 5a² − 2ab + 3a² + ab − b².

示例B：化简 5a² − 2ab + 3a² + ab − b²。

Group like terms: (5a² + 3a²) + (−2ab + ab) − b².

合并同类项：(5a² + 3a²) + (−2ab + ab) − b²。

Compute: 8a² − ab − b². This expression cannot be simplified further.

计算后得到：8a² − ab − b²。这个表达式无法继续化简。

10. Practice Problems | 练习题目

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

请独立尝试以下题目，再对照下方的答案。

Problem / 题目	Answer / 答案
1. 7x + 2x	9x
2. 5y − 8y + 3y	0y = 0
3. 4a + 3b − a + 7b	3a + 10b
4. 2(x + 4) + 3x	5x + 8
5. 6m² − 2m + 3m² + 5m	9m² + 3m

11. Real-World Applications | 实际应用

Combining like terms is not just an abstract exercise. It appears in many real-world situations, such as calculating perimeter or area with algebraic expressions.

合并同类项并不只是抽象练习。它在许多现实情境中都会出现，例如用代数表达式计算周长或面积。

Example: A rectangle has sides (2x + 1) and (x + 3). What is its perimeter?

示例：一个长方形的边长分别为 (2x + 1) 和 (x + 3)。它的周长是多少？

Perimeter = 2(2x + 1) + 2(x + 3). Expand: 4x + 2 + 2x + 6. Combine like terms: (4x + 2x) + (2 + 6) = 6x + 8.

周长 = 2(2x + 1) + 2(x + 3)。展开：4x + 2 + 2x + 6。合并同类项：(4x + 2x) + (2 + 6) = 6x + 8。

This technique is also used in physics when adding forces, in economics when combining costs, and in computer science when simplifying algorithms. A strong foundation in combining like terms will benefit you across all STEM subjects.

这种技巧也用于物理中力的合成、经济学中成本的汇总，以及计算机科学中算法的简化。牢固掌握合并同类项将对你学习所有理工科科目都有帮助。

12. Summary | 总结

To combine like terms, follow these simple rules: identify terms with the same variable and same power; add or subtract their coefficients; keep the variable part unchanged. Always expand brackets first if needed, and be careful with signs and invisible coefficients.

合并同类项遵循以下简单规则：找出变量相同且指数相同的项；对它们的系数做加减；变量部分保持不变。如果需要，先展开括号，并注意符号和隐含系数。

Remember that like terms are the only terms that can be combined. Unlike terms must remain separate. With practice, you will be able to simplify algebraic expressions quickly and accurately, which is a skill you will use again and again in IGCSE Mathematics and beyond.

请记住：只有同类项才能合并。非同类项必须保留。通过练习，你将能够快速、准确地化简代数表达式，这是你在IGCSE数学及以后的数学学习中会反复使用的一项技能。

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Combining Like Terms | 合并同类项