Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain multiple terms that can be simplified. Collecting like terms is the process of combining terms that have the same variable raised to the same power, making expressions cleaner and easier to work with.

在代数中，表达式通常包含多个可以简化的项。合并同类项就是把具有相同变量且变量指数相同的项合并在一起的过程，它使表达式更加简洁，也更容易进行后续运算。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable parts, including the same powers. For example, 3x and 5x are like terms because both contain the variable x raised to the first power. The coefficient — the number in front of the variable — can be different.

同类项是指变量部分完全相同（包括相同的指数）的项。例如，3x 和 5x 是同类项，因为两者都包含一次方的变量 x。而系数——变量前面的数字——可以不同。

Like terms: 2a and 7a, 4x² and −3x², xy and 6xy
同类项：2a 与 7a，4x² 与 −3x²，xy 与 6xy
Not like terms: 2a and 2b, 4x² and 4x, xy and x²y
非同类项：2a 与 2b，4x² 与 4x，xy 与 x²y

Notice that the variable must match exactly, including its exponent. The term 4x² cannot be combined with 4x because one is squared and the other is not.

请注意，变量必须完全一致，包括其指数。4x² 不能与 4x 合并，因为一个是平方项，另一个不是。

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

Simplifying expressions by collecting like terms is essential for several reasons. First, it reduces the complexity of an expression, making it easier to substitute values or solve equations. Second, it reveals the structure of the expression, allowing us to see relationships more clearly.

通过合并同类项来化简表达式至关重要，原因有以下几点：首先，它降低了表达式的复杂程度，使代入数值或解方程更加简便；其次，它揭示了表达式的结构，让我们能更清楚地看到各项之间的关系。

For instance, the expression 7x + 3x − 2x can be simplified to 8x in one step. This simplified form is much easier to evaluate when x has a specific value, such as x = 5, giving 8 × 5 = 40.

例如，表达式 7x + 3x − 2x 可以一步化简为 8x。这种简化形式在 x 取特定值时更容易求值，比如 x = 5 时，结果为 8 × 5 = 40。

In examinations, un-simplified answers are often marked as incorrect or partially incorrect. Therefore, mastering this skill is fundamental for success in IGCSE Mathematics.

在考试中，未化简的答案常常被判定为错误或部分错误。因此，掌握这一技能是 IGCSE 数学取得好成绩的基础。

3. The Basic Rule of Adding and Subtracting Coefficients | 合并同类项的基本法则

To collect like terms, simply add or subtract their coefficients while keeping the variable part unchanged. For example, 4x + 3x = 7x because 4 + 3 = 7. Similarly, 9y − 5y = 4y because 9 − 5 = 4.

合并同类项时，只需对系数进行加减运算，变量部分保持不变。例如，4x + 3x = 7x，因为 4 + 3 = 7；同理，9y − 5y = 4y，因为 9 − 5 = 4。

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

a x − b x = (a − b)x

When there is no coefficient written in front of a variable, the coefficient is understood to be 1. For example, x + x = 2x, and 3x + x = 4x.

当变量前面没有写系数时，默认系数为 1。例如，x + x = 2x，而 3x + x = 4x。

Be careful with negative signs: −2a + 5a = 3a, and −3b − 4b = −7b. Always apply the sign in front of each term when collecting.

要特别注意负号：−2a + 5a = 3a，−3b − 4b = −7b。合并时一定要带上每一项前面的符号。

4. Collecting Terms with Different Variables | 合并含有不同变量的项

Terms with different variables cannot be combined. For example, in the expression 5a + 3b − 2a + 4b, we can collect a-terms together and b-terms separately.

含有不同变量的项不能合并。例如，在表达式 5a + 3b − 2a + 4b 中，我们可以分别把含 a 的项和含 b 的项各自合并。

5a − 2a = 3a

3b + 4b = 7b

Therefore: 5a + 3b − 2a + 4b = 3a + 7b

因此：5a + 3b − 2a + 4b = 3a + 7b

It is conventional to write terms in alphabetical order, or to group terms with the same variable together. This makes the final expression clearer and more standardised.

通常按字母顺序排列各项，或将含相同变量的项归组排列。这样最终表达式更清晰、更规范。

Also note that x² and x are not like terms. The expression x² + x cannot be simplified further because the powers differ. However, x² + 3x² = 4x² is valid.

还需注意，x² 和 x 不是同类项。表达式 x² + x 不能进一步化简，因为指数不同。但 x² + 3x² = 4x² 是成立的。

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

Terms such as 2xy, 5xy and −3xy are like terms because they all contain the product xy. We can combine them: 2xy + 5xy − 3xy = 4xy. The variable part xy remains unchanged.

像 2xy、5xy 和 −3xy 这样的项是同类项，因为它们都包含乘积 xy。我们可以合并它们：2xy + 5xy − 3xy = 4xy。变量部分 xy 保持不变。

However, xy and x²y are not like terms, because the powers of x differ. Similarly, xy and xyz are not like terms because one contains z and the other does not.

然而，xy 与 x²y 不是同类项，因为 x 的指数不同。同理，xy 与 xyz 也不是同类项，因为其中一个含 z，另一个不含。

3pq − 2pq + 7pq = (3 − 2 + 7)pq = 8pq

When multiple variables are involved, always compare every variable and its exponent to determine whether terms are like terms.

当涉及多个变量时，必须逐一比较每个变量及其指数，才能判断是否为同类项。

6. Collecting Terms with Indices | 合并含指数的同类项

Terms with the same variable but different exponents are not like terms. For instance, x² and x³ cannot be combined. Only terms with exactly the same exponent can be collected.

变量相同但指数不同的项不是同类项。例如，x² 与 x³ 不能合并。只有当指数完全相同时才能合并。

2x² + 5x² = 7x² ✓
2x² + 5x² = 7x⁴ ✗ (incorrect — powers do not add)
2x² + 5x² = 7x⁴ ✗（错误——指数不能相加）

This is a very common mistake among students. Remember: when adding or subtracting like terms, the exponent never changes; only the coefficient changes.

这是学生中最常见的错误之一。请记住：在加减同类项时，指数永远不会改变，只有系数发生变化。

4x³ − 2x³ + x³ = (4 − 2 + 1)x³ = 3x³

The same principle applies to terms with numbers and variables combined, such as 2x²y and −5x²y, which can be combined to give −3x²y.

同样的原则也适用于数字与变量结合的项，例如 2x²y 和 −5x²y 可以合并为 −3x²y。

7. Simplifying Expressions with Brackets | 先去括号再合并

When an expression contains brackets, we must first expand the brackets using the distributive law, and then collect like terms. This is a two-step process.

当表达式包含括号时，必须先利用分配律展开括号，然后才能合并同类项。这是一个两步过程。

3(x + 2) + 2(x + 1) = 3x + 6 + 2x + 2 = 5x + 8

In this example, we first multiplied each term inside the brackets by the number outside. Then we collected the x-terms (3x + 2x = 5x) and the constant terms (6 + 2 = 8).

在这个例子中，我们先用括号外的数乘以括号内的每一项，然后分别合并含 x 的项（3x + 2x = 5x）和常数项（6 + 2 = 8）。

When the term outside the bracket is negative, be especially careful with signs. For instance, 4(2x − 3) − 2(3x − 1) requires careful sign handling.

当括号外的数是负数时，要特别注意符号。例如 4(2x − 3) − 2(3x − 1) 就需要小心处理符号。

4(2x − 3) − 2(3x − 1) = 8x − 12 − 6x + 2 = 2x − 10

Notice how −2(3x − 1) becomes −6x + 2: the negative sign multiplies both terms inside the bracket.

注意 −2(3x − 1) 展开为 −6x + 2：负号同时乘以括号内的每一项。

8. Common Mistakes and How to Avoid Them | 常见错误及防范方法

Students frequently make certain predictable errors when collecting like terms. Being aware of these pitfalls is the first step toward avoiding them.

学生在合并同类项时经常犯一些可预见的错误。了解这些陷阱是避免它们的第一步。

Mistake \| 错误	Correct \| 正确	Explanation \| 解释
x² + x² = x⁴	x² + x² = 2x²	Exponents do not change when adding \| 加法中指数不变
3a + 2b = 5ab	3a + 2b (cannot simplify)	Different variables — cannot combine \| 变量不同，不能合并
5 − 2(x + 3) = 3(x + 3)	5 − 2x − 6 = −2x − 1	Brackets must be expanded first \| 必须先展开括号

To avoid these errors, always follow this order: first expand brackets, then identify like terms, and finally combine coefficients carefully.

要避免这些错误，请始终遵循以下顺序：先展开括号，再识别同类项，最后仔细合并系数。

Another common error is forgetting that a term like −x has a coefficient of −1. So −x + 2x = x, not 0, since −1 + 2 = 1.

另一个常见错误是忘记像 −x 这样的项系数为 −1。因此 −x + 2x = x，而不是 0，因为 −1 + 2 = 1。

9. Collecting Like Terms in Geometry Applications | 合并同类项在几何中的应用

Perimeter calculations often require collecting like terms. For a rectangle with length (2x + 5) and width (x − 2), the perimeter P is given by 2(length + width).

周长计算经常需要合并同类项。对于一个长为 (2x + 5)、宽为 (x − 2) 的矩形，周长 P 由 2（长 + 宽）给出。

P = 2[(2x + 5) + (x − 2)] = 2[3x + 3] = 6x + 6

The simplification step 2x + 5 + x − 2 = 3x + 3 requires collecting the x-terms (2x + x = 3x) and the constant terms (5 − 2 = 3).

化简步骤 2x + 5 + x − 2 = 3x + 3 需要合并含 x 的项（2x + x = 3x）和常数项（5 − 2 = 3）。

Similarly, if a triangle has sides measuring (3a − 1), (2a + 4) and (5a − 2), the perimeter is the sum of all three sides.

同理，如果一个三角形的三条边分别为 (3a − 1)、(2a + 4) 和 (5a − 2)，则周长等于三边之和。

P = (3a − 1) + (2a + 4) + (5a − 2) = 10a + 1

In this case, the coefficient of a is 3 + 2 + 5 = 10, and the constant term is −1 + 4 − 2 = 1.

在此例中，a 的系数为 3 + 2 + 5 = 10，常数项为 −1 + 4 − 2 = 1。

10. Exam-Style Worked Examples | 考试题型范例

Let us work through some typical IGCSE questions step by step to consolidate the concepts covered in this article.

下面我们逐步解答一些典型的 IGCSE 题目，以巩固本文所学的概念。

Example 1 | 例 1: Simplify 7x + 3y − 2x + 5y

解： First collect the x-terms: 7x − 2x = 5x. Then collect the y-terms: 3y + 5y = 8y. Therefore the answer is 5x + 8y.

例 1： 化简 7x + 3y − 2x + 5y

解答： 先合并含 x 的项：7x − 2x = 5x。再合并含 y 的项：3y + 5y = 8y。因此答案为 5x + 8y。

Example 2 | 例 2: Simplify 4(2x − 3) + 3(x + 2)

Expand the brackets: 8x − 12 + 3x + 6. Collect like terms: 8x + 3x = 11x and −12 + 6 = −6. The simplified expression is 11x − 6.

例 2： 化简 4(2x − 3) + 3(x + 2)

展开括号：8x − 12 + 3x + 6。合并同类项：8x + 3x = 11x，−12 + 6 = −6。化简结果为 11x − 6。

Example 3 | 例 3: Simplify 2x² + 3x − x² + 5x

Collect the x² terms: 2x² − x² = x². Collect the x terms: 3x + 5x = 8x. The answer is x² + 8x.

例 3： 化简 2x² + 3x − x² + 5x

合并 x² 项：2x² − x² = x²。合并 x 项：3x + 5x = 8x。答案为 x² + 8x。

11. Practice Questions | 练习题目

Try these problems on your own before checking the answers. They cover all the techniques discussed in this article.

请先独立尝试以下题目，再对照答案。它们涵盖了本文讨论的所有技巧。

Question 1 | 第 1 题: Simplify 6a + 2b − 3a + 4b

Question 2 | 第 2 题: Simplify 5x² + 3x − 2x² + x

Question 3 | 第 3 题: Simplify 3(2x + 1) + 2(4x − 3)

Question 4 | 第 4 题: Simplify x² + 2xy + 3y² − 2x² + xy − y²

Answers | 答案:

1. 3a + 6b 2. 3x² + 4x 3. 14x − 3 4. −x² + 3xy + 2y²

For question 4, collect x²-terms (1 − 2 = −1), xy-terms (2 + 1 = 3) and y²-terms (3 − 1 = 2), giving −x² + 3xy + 2y².

对于第 4 题，分别合并 x² 项（1 − 2 = −1）、xy 项（2 + 1 = 3）和 y² 项（3 − 1 = 2），得到 −x² + 3xy + 2y²。

12. Summary and Key Takeaways | 总结与要点

Collecting like terms is one of the most fundamental skills in algebra. It appears in almost every topic, from solving equations to expanding brackets, from factorisation to coordinate geometry.

合并同类项是代数中最基础的技能之一。它几乎出现在每一个主题中，从解方程到展开括号，从因式分解到坐标几何。

Like terms must have identical variable parts, including exponents | 同类项的变量部分必须完全相同，包括指数
Only coefficients are added or subtracted — variables and exponents stay unchanged | 只有系数进行加减——变量和指数保持不变
Terms with different variables or different exponents cannot be combined | 变量不同或指数不同的项不能合并
Always expand brackets before collecting like terms | 合并同类项之前一定要先展开括号
Pay careful attention to negative signs | 要特别注意负号

With consistent practice, collecting like terms will become second nature, allowing you to manipulate algebraic expressions quickly and accurately in both examinations and future mathematical studies.

通过持续练习，合并同类项会成为你的本能反应，使你在考试和未来的数学学习中都能快速、准确地处理代数表达式。

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Collecting Like Terms | 合并同类项