Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

Combining like terms is one of the most fundamental skills in algebra. It refers to the process of simplifying algebraic expressions by adding or subtracting terms that have exactly the same variable parts. This skill underpins nearly every topic in IGCSE Mathematics, from solving equations to manipulating quadratic expressions.

合并同类项是代数中最基础的技能之一。它指的是通过加减那些变量部分完全相同的项来化简代数表达式的过程。这项技能贯穿IGCSE数学的几乎所有主题，从解方程到二次表达式的运算都离不开它。

1. What Are “Terms” and “Like Terms”? | 什么是”项”和”同类项”？

In algebra, the parts of an expression that are separated by + or − signs are called “terms”. For instance, in the expression 3x + 5y − 2x, there are three terms: 3x, 5y, and −2x. Each term consists of a coefficient (the number in front) and a variable part.

在代数中，用”+”或”−”号分隔的表达式的各个部分称为”项”。例如，表达式 3x + 5y − 2x 中有三项：3x、5y 和 −2x。每一项都由系数（前面的数字）和变量部分组成。

Like terms are terms that have exactly the same variable part, including the same powers. For example, 3x and −2x are like terms because both have the variable x to the first power. However, 3x and 3y are NOT like terms, because x and y are different variables.

同类项是指变量部分完全相同（包括相同的幂次）的项。例如，3x 和 −2x 是同类项，因为它们的变量都是 x 的一次幂。然而，3x 和 3y 不是同类项，因为 x 和 y 是不同的变量。

5x² and −2x² are like terms (both contain x²).
5x² and −2x are NOT like terms (the powers differ).
7xy and 3yx are like terms (xy = yx, commutative property).

5x² 和 −2x² 是同类项（都含有 x²）。
5x² 和 −2x 不是同类项（幂次不同）。
7xy 和 3yx 是同类项（xy = yx，乘法交换律）。

2. The Coefficient | 系数

The coefficient is the numerical factor in a term. In the term 7x, the coefficient is 7. If a term has no number written in front, such as x or −y, the coefficient is understood to be 1 or −1 respectively. Recognising coefficients is essential because when we combine like terms, we only add or subtract the coefficients while keeping the variable part unchanged.

系数是项中的数字因子。在 7x 项中，系数是 7。如果一项前面没有写数字，例如 x 或 −y，其系数分别默认为 1 或 −1。识别系数至关重要，因为合并同类项时，我们只对系数进行加减，而保持变量部分不变。

Like terms combine by adding or subtracting coefficients: ax + bx = (a + b)x

同类项通过系数加减合并：ax + bx = (a + b)x

3. Combining Like Terms with Addition | 加法合并同类项

When simplifying expressions, the first step is always to identify and group like terms. For example, simplify 3x + 2x. Both terms contain x, so we add the coefficients: 3 + 2 = 5, giving 5x. The variable part x stays the same.

在化简表达式时，第一步永远是识别并分组同类项。例如，化简 3x + 2x。两项都含有 x，所以我们相加系数：3 + 2 = 5，得到 5x。变量部分 x 保持不变。

Example 1: Simplify 4a + 7a = (4 + 7)a = 11a

例1：化简 4a + 7a = (4 + 7)a = 11a

If an expression contains several terms, gather all like terms together before combining. Simplify 5x + 3y − 2x + y. Group x-terms: 5x − 2x = 3x. Group y-terms: 3y + y = 4y. The simplified expression is 3x + 4y.

如果表达式中包含多项，先收集所有同类项再进行合并。化简 5x + 3y − 2x + y。将 x 项分组：5x − 2x = 3x。将 y 项分组：3y + y = 4y。化简后的表达式为 3x + 4y。

4. Combining Like Terms with Subtraction | 减法合并同类项

Subtraction follows the same principle. Simplify 9p − 4p. Since both terms contain p, subtract coefficients: 9 − 4 = 5. The result is 5p. Be careful with negative terms: consider 2x − 7x. The coefficients give 2 − 7 = −5, so the answer is −5x.

减法遵循同样的原则。化简 9p − 4p。由于两项都含有 p，系数相减：9 − 4 = 5。结果为 5p。注意负项：考虑 2x − 7x，系数为 2 − 7 = −5，所以答案为 −5x。

Example 2: Simplify 6m − 10m = (6 − 10)m = −4m

例2：化简 6m − 10m = (6 − 10)m = −4m

When there are multiple terms with different signs, treat each term’s sign as part of the term. Simplify 4x − 2y + 3x − 5y. x-terms: 4x + 3x = 7x. y-terms: −2y − 5y = −7y. Result: 7x − 7y.

当多个项带有不同符号时，将每项的符号视为该项的一部分。化简 4x − 2y + 3x − 5y。x 项：4x + 3x = 7x。y 项：−2y − 5y = −7y。结果：7x − 7y。

5. Different Powers Cannot Be Combined | 不同幂次不能合并

A very common error is attempting to combine terms such as x² and x. These are not like terms because the powers differ. The expression x² + x cannot be simplified further. Similarly, x² + x³ cannot be combined to make x⁵ or x⁶ — each power must be treated separately.

一个非常常见的错误是试图合并 x² 和 x 这类项。它们不是同类项，因为幂次不同。表达式 x² + x 不能再化简。同样，x² + x³ 不能合并成 x⁵ 或 x⁶——每个幂次必须单独处理。

Correct: 3x² + 5x² = 8x² | Incorrect: 3x² + 5x = 8x² (or 8x)

正确：3x² + 5x² = 8x² | 错误：3x² + 5x = 8x²（或 8x）

Think of x² and x as different objects — like apples and oranges. You can add 3 apples and 5 apples to get 8 apples, but you cannot add 3 apples and 5 oranges to get 8 “apple-oranges”. The same logic applies to different powers.

可以把 x² 和 x 视为不同的物体——就像苹果和橙子。3 个苹果加 5 个苹果得到 8 个苹果，但 3 个苹果加 5 个橙子不能得到 8 个”苹果橙子”。不同的幂次也是同样的道理。

6. Simplifying Expressions with Multiple Variables | 多重变量表达式的化简

When an expression contains several different variables, group each type of like term separately. Simplify 2ab + 3a − ab + b. The term 2ab and −ab are like terms: 2ab − ab = ab. The terms 3a and b have no like terms, so they remain. The result is ab + 3a + b — note we commonly write ab before 3a by convention, but any order is acceptable.

当表达式包含多个不同变量时，分别对每种同类型的项进行分组。化简 2ab + 3a − ab + b。项 2ab 和 −ab 是同类项：2ab − ab = ab。项 3a 和 b 没有同类项，保持不变。结果为 ab + 3a + b——注意按照惯例我们通常把 ab 写在 3a 前面，但实际上任何顺序都可以。

Example 3: Simplify 5x + 3y − 2x + 7y − x = (5 − 2 − 1)x + (3 + 7)y = 2x + 10y

例3：化简 5x + 3y − 2x + 7y − x = (5 − 2 − 1)x + (3 + 7)y = 2x + 10y

It is also important to note that the variable order within a term does not change “likeness”. For example, 2xy and 5yx are like terms because xy = yx by the commutative property of multiplication. Similarly, 3abc and −4bca are like terms.

还需要注意，项内变量的顺序不改变”同类性”。例如，2xy 和 5yx 是同类项，因为根据乘法交换律 xy = yx。同样，3abc 和 −4bca 也是同类项。

7. Combining After Expanding Brackets | 去括号后再合并

In IGCSE exams, brackets often appear alongside like terms. The rule is: always expand brackets first, then collect like terms. Simplify 3(x + 2) + 4x. First expand: 3(x + 2) = 3x + 6. The expression becomes 3x + 6 + 4x. Now combine like terms: 3x + 4x = 7x. The final answer is 7x + 6.

在IGCSE考试中，括号经常与同类项一起出现。规则是：先去括号，再合并同类项。化简 3(x + 2) + 4x。先展开：3(x + 2) = 3x + 6。表达式变为 3x + 6 + 4x。然后合并同类项：3x + 4x = 7x。最终答案为 7x + 6。

Example 4: Simplify 2(3x − 1) − (2x + 5) = 6x − 2 − 2x − 5 = 4x − 7

例4：化简 2(3x − 1) − (2x + 5) = 6x − 2 − 2x − 5 = 4x − 7

Be especially careful when subtracting a bracket: the minus sign in front of the bracket changes the sign of every term inside. That is why −(2x + 5) becomes −2x − 5.

特别注意减去括号时：括号前的负号会改变括号内每一项的符号。这就是为什么 −(2x + 5) 变成 −2x − 5。

8. Combining Like Terms with Fractions | 含分数的同类项合并

When coefficients are fractions, the process remains identical — combine fractions using common denominators. Simplify ½x + ⅓x. Since both terms are like terms, add the fractions: ½ + ⅓ = 3⁄6 + 2⁄6 = 5⁄6. Therefore, ½x + ⅓x = 5⁄6x.

当系数为分数时，过程完全相同——用公共分母对分数进行加减。化简 ½x + ⅓x。由于两项是同类项，将分数相加：½ + ⅓ = 3⁄6 + 2⁄6 = 5⁄6。因此，½x + ⅓x = 5⁄6x。

Example 5: Simplify ¾y − ¼y = (¾ − ¼)y = ½y

例5：化简 ¾y − ¼y = (¾ − ¼)y = ½y

Decimal coefficients work the same way too: 0.5x + 0.25x = 0.75x. No matter what form the coefficient takes, the rule is always to combine only the numbers, leaving the variable untouched.

小数系数也同样适用：0.5x + 0.25x = 0.75x。无论系数是什么形式，规则始终是只合并数字部分，保持变量不变。

9. Common Mistakes to Avoid | 常见错误警示

The most frequent errors students make in combining like terms are incorrect sign handling, combining unlike terms, and overlooking hidden coefficients. Below is a table of common errors and their corrections:

学生在合并同类项时最常见的错误包括符号处理不当、合并非同类项以及忽略隐含系数。下表列出了常见错误及其修正：

Expression	Common Mistake	Correct Answer
2x − 3x	−5x (adding instead of subtracting)	−x
x² + x	x³ (adding powers)	x² + x (cannot simplify further)
3x + 5y	8xy (treating different variables as same)	3x + 5y (cannot simplify further)
2a + a	2a (forgetting a = 1a)	3a

Equation | 表达式 | 常见错误 | 正确答案

10. Real Exam Question Walkthroughs | 真题实战演练

Let us now work through questions styled after IGCSE past papers. While these are not from a specific exam board paper, they reflect the typical difficulty and format you can expect.

现在让我们练习几道模仿IGCSE真题风格的问题。这些题目并非来自某个特定考试局的试卷，但反映了你可能会遇到的典型难度和格式。

Question 1: Subtract 5y + 3 from 8y + 2. | 问题1：从 8y + 2 中减去 5y + 3。

Solution: (8y + 2) − (5y + 3) = 8y + 2 − 5y − 3 = (8 − 5)y + (2 − 3) = 3y − 1.

解答：(8y + 2) − (5y + 3) = 8y + 2 − 5y − 3 = (8 − 5)y + (2 − 3) = 3y − 1。

Question 2: Simplify 7a + 5b − 2a + 3b − 4a. | 问题2：化简 7a + 5b − 2a + 3b − 4a。

Solution: a-terms: 7a − 2a − 4a = 1a = a. b-terms: 5b + 3b = 8b. Answer: a + 8b.

解答：a 项：7a − 2a − 4a = 1a = a。b 项：5b + 3b = 8b。答案：a + 8b。

Question 3: Find the perimeter of a rectangle with length 3x + 2 and width x − 1. | 问题3：求长为 3x + 2、宽为 x − 1 的矩形的周长。

Solution: Perimeter = 2 × (length + width) = 2 × [(3x + 2) + (x − 1)] = 2 × (4x + 1) = 8x + 2.

解答：周长 = 2 × (长 + 宽) = 2 × [(3x + 2) + (x − 1)] = 2 × (4x + 1) = 8x + 2。

11. Combining Like Terms in Quadratic Expressions | 二次表达式中的同类项合并

In IGCSE mathematics, you will frequently encounter quadratic expressions such as 3x² + 5x − 2x² + 4x. To simplify, group the x² terms together and the x terms together: (3x² − 2x²) + (5x + 4x) = x² + 9x. If there is a constant term (a number with no variable), combine it separately.

在IGCSE数学中，你经常会遇到二次表达式，如 3x² + 5x − 2x² + 4x。化简时，将 x² 项分在一起，将 x 项分在一起：(3x² − 2x²) + (5x + 4x) = x² + 9x。如果存在常数项（没有变量的数字），将其单独合并。

Example 6: Simplify 4x² − 3x + 5 + 2x² + 7x − 1 = (4 + 2)x² + (−3 + 7)x + (5 − 1) = 6x² + 4x + 4

例6：化简 4x² − 3x + 5 + 2x² + 7x − 1 = (4 + 2)x² + (−3 + 7)x + (5 − 1) = 6x² + 4x + 4

This skill is essential for solving quadratic equations later: before you can factorise or use the quadratic formula, the equation must first be simplified into the form ax² + bx + c = 0. Without a solid grasp of combining like terms, this preliminary step will trip you up.

这项技能对于之后解二次方程至关重要：在因式分解或使用二次公式之前，必须先将方程化简为 ax² + bx + c = 0 的形式。如果不扎实掌握合并同类项，这个预备步骤就会让你出错。

12. Exam Tips and Quick Checks | 考试技巧与快速检验

Here are some final tips to help you excel in your IGCSE exams when it comes to combining like terms:

以下是一些最后的技巧，帮助你在IGCSE考试中更好地应对合并同类项的问题：

Always scan the expression first and underline or circle each family of like terms with a different symbol before combining.
Keep the sign (+ or −) with its term — never detach it.
Remember the order: expand brackets → combine like terms → arrange final answer in descending powers.

先通览表达式，用不同符号在每类同类项下方划线或画圈，然后再合并。
始终将符号（+ 或 −）与其项放在一起——切勿分离。
记住顺序：去括号 → 合并同类项 → 按降幂排列最终答案。

Finally, always check your work by substituting a simple value for the variable, such as x = 1, into both the original expression and your simplified answer. If they give the same result, your simplification is correct. This quick method takes seconds but catches many careless mistakes.

最后，始终通过代入一个简单的变量值来检查你的工作，例如 x = 1，分别代入原表达式和化简后的答案。如果结果相同，说明化简正确。这个快速方法只需几秒钟，却能捕捉到许多粗心错误。

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Combining Like Terms | 合并同类项