Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Collecting like terms is one of the most fundamental skills in algebra. It allows us to simplify algebraic expressions by combining terms that share the same variable and the same power. This skill appears in nearly every IGCSE Mathematics exam paper, often as part of a larger problem.

合并同类项是代数中最基础的技能之一。它通过合并具有相同变量和相同幂次的项来简化代数表达式。这一技能几乎出现在每一份 IGCSE 数学试卷中，通常是更大题目的一部分。

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

An algebraic expression is made up of terms. A term is a number, a variable, or a product of numbers and variables. For example, in the expression 3x + 5y − 2, there are three terms: 3x, 5y, and −2.

代数表达式由项组成。项是一个数字、一个变量、或数字与变量的乘积。例如，在表达式 3x + 5y − 2 中，有三个项：3x、5y 和 −2。

Like terms are terms that have exactly the same variables raised to exactly the same powers. Only the numerical coefficients may differ.

同类项是指具有完全相同变量且变量幂次完全相同的项。只有数字系数可以不同。

Examples of like terms: 2x and 5x are like terms; 4x² and 7x² are like terms.

同类项举例：2x 和 5x 是同类项；4x² 和 7x² 是同类项。

2. Why Can We Only Combine Like Terms? | 为什么只能合并同类项

Algebra is a generalised form of arithmetic. In arithmetic, 2 apples + 3 apples = 5 apples. Likewise, 2x + 3x = 5x because x represents an unknown quantity. However, 2 apples + 3 oranges cannot be simplified to 5 apple-oranges. Similarly, x + x² cannot be simplified because x and x² represent different quantities.

代数是算术的推广形式。在算术中，2个苹果 + 3个苹果 = 5个苹果。同理，2x + 3x = 5x，因为 x 代表一个未知的数量。然而，2个苹果 + 3个橙子不能被简化为5个”苹果橙子”。同样，x + x² 不能被简化，因为 x 和 x² 代表不同的量。

For example, if x = 2, then x + x² = 2 + 4 = 6, but 2x = 4 and x² = 4. The two terms are not the same type of quantity, so they cannot be added together in a simplified form.

例如，如果 x = 2，那么 x + x² = 2 + 4 = 6，但是 2x = 4，x² = 4。这两个项不是同一种量，因此不能以简化形式相加。

3. The Basic Rule: Add or Subtract Coefficients | 基本规则：系数的加减

To collect like terms, keep the variable part unchanged and add or subtract the numerical coefficients. For example:

合并同类项时，保持变量部分不变，对数字系数进行加减运算。例如：

4x + 3x = (4 + 3)x = 7x

4x − 3x = (4 − 3)x = 1x = x

Remember that when a coefficient is not written, it is understood to be 1. So x means 1x, and −y means −1y.

请记住，当系数没有写出时，它默认为 1。所以 x 表示 1x，−y 表示 −1y。

4. Distinguishing Between Different Kinds of Terms | 区分不同类型的项

When simplifying an expression, you must classify each term carefully before combining. The table below shows how to classify terms in the expression 3x² − 5x + 2x² + 7 − x − 4.

在简化表达式时，必须先仔细对每一项进行分类，然后再进行合并。下表展示了如何对表达式 3x² − 5x + 2x² + 7 − x − 4 中的各项进行分类。

Term \| 项	Type \| 类型
3x²	x² term \| x² 项
−5x and −x	x terms \| x 项
7 and −4	constant terms \| 常数项
2x²	x² term \| x² 项

3x² + 2x² = 5x²; −5x − x = −6x; 7 − 4 = 3

3x² + 2x² = 5x²；−5x − x = −6x；7 − 4 = 3

Therefore, the expression simplifies to 5x² − 6x + 3.

因此，表达式简化为 5x² − 6x + 3。

5. Worked Example with Negative Coefficients | 含负系数的例题

Negative coefficients often cause errors in exams. Consider the expression −7a + 3b − 2a − 5b. Identify like terms: the a terms are −7a and −2a; the b terms are 3b and −5b.

负系数在考试中经常导致错误。考虑表达式 −7a + 3b − 2a − 5b。识别同类项：a 项是 −7a 和 −2a；b 项是 3b 和 −5b。

−7a − 2a = −9a; 3b − 5b = −2b

−7a − 2a = −9a；3b − 5b = −2b

So the simplified expression is −9a − 2b. Notice that the sign belongs to the term that follows it. When you rearrange terms, carry the sign with the term.

因此简化后的表达式是 −9a − 2b。注意符号属于它后面的项。当你重新排列各项时，符号要跟着项一起移动。

6. Simplifying with Different Powers of the Same Variable | 同一变量不同幂次的化简

When a variable appears with different powers, each power group must be collected separately. For example, in 4x² + 3x + 5x² − 2x, the x² terms are 4x² and 5x², while the x terms are 3x and −2x.

当同一变量以不同幂次出现时，每个幂次组必须分别合并。例如，在 4x² + 3x + 5x² − 2x 中，x² 项是 4x² 和 5x²，而 x 项是 3x 和 −2x。

4x² + 5x² = 9x²; 3x − 2x = x

4x² + 5x² = 9x²；3x − 2x = x

The answer is 9x² + x. Do NOT combine 9x² with x, because x² and x are not like terms. This is one of the most common mistakes in IGCSE algebra questions.

答案是 9x² + x。不要把 9x² 与 x 合并，因为 x² 和 x 不是同类项。这是 IGCSE 代数题目中最常见的错误之一。

7. Collecting Terms with Multiple Variables | 含多个变量的合并同类项

Terms with different variable combinations are not like terms. For instance, xy and x are not like terms; x²y and xy² are not like terms either. The order of variables within a term does not matter for classification, but the set of variables and powers must match exactly.

具有不同变量组合的项不是同类项。例如，xy 和 x 不是同类项；x²y 和 xy² 也不是同类项。项内变量的顺序不影响分类，但变量集合及幂次必须完全匹配。

3xy + 5xy = 8xy; but 3xy + 5x = 3xy + 5x (cannot be simplified further)

3xy + 5xy = 8xy；但 3xy + 5x = 3xy + 5x（不能再简化）

Also note that 2xy and 2yx are like terms because xy = yx by the commutative property of multiplication.

同时注意，2xy 和 2yx 是同类项，因为根据乘法交换律 xy = yx。

8. Expanding Brackets Before Collecting | 先去括号再合并

Often, expressions contain brackets that must be expanded first. Use the distributive law: a(b + c) = ab + ac. After expanding, collect like terms.

通常，表达式中包含括号，必须先展开。使用分配律：a(b + c) = ab + ac。展开后再合并同类项。

Example | 例题: Simplify 3(x + 2) + 2(x − 1).

例题：化简 3(x + 2) + 2(x − 1)。

Expand: 3(x + 2) = 3x + 6, and 2(x − 1) = 2x − 2. Then combine: 3x + 2x = 5x, and 6 − 2 = 4. The answer is 5x + 4.

展开：3(x + 2) = 3x + 6，2(x − 1) = 2x − 2。然后合并：3x + 2x = 5x，6 − 2 = 4。答案为 5x + 4。

9. Handling Negative Signs Before Brackets | 处理括号前的负号

When a negative sign appears before a bracket, it multiplies every term inside by −1. This is a common source of sign errors.

当括号前出现负号时，它表示将括号内的每一项都乘以 −1。这是符号错误的常见来源。

Example | 例题: Simplify 5x − (2x − 3).

例题：化简 5x − (2x − 3)。

Expand: −(2x − 3) = −2x + 3. Now collect: 5x − 2x = 3x, and the constant is +3. The answer is 3x + 3.

展开：−(2x − 3) = −2x + 3。现在合并：5x − 2x = 3x，常数项为 +3。答案为 3x + 3。

10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Mistake 1: Combining x with x². Always check the powers before combining terms.

错误一：将 x 与 x² 合并。合并前务必检查幂次。
Mistake 2: Dropping negative signs. In the expression 2a − 3a − b, the result is −a − b, not a − b. Keep the sign with its term.

错误二：丢失负号。在表达式 2a − 3a − b 中，结果为 −a − b，而不是 a − b。符号要跟随它的项。
Mistake 3: Treating x and xy as like terms. They are not the same type of term, so they cannot be combined.

错误三：将 x 和 xy 当作同类项。它们不是同一种项，因此不能合并。
Mistake 4: Incorrectly expanding a bracket. When expanding a(b + c), multiply a by BOTH b and c, not just the first term.

错误四：错误地展开括号。展开 a(b + c) 时，a 要乘以 b 和 c 两项，而不仅仅是第一项。

11. Works in Context: Perimeter and Word Problems | 实际应用：周长与文字题

Collecting like terms is frequently tested in problem-solving contexts. For example, the perimeter of a rectangle with length 2x + 3 and width x − 1 can be expressed as follows:

合并同类项经常在解决问题的场景中考查。例如，长为 2x + 3、宽为 x − 1 的矩形，其周长可表示如下：

Perimeter = 2(2x + 3) + 2(x − 1) = 4x + 6 + 2x − 2 = 6x + 4

周长 = 2(2x + 3) + 2(x − 1) = 4x + 6 + 2x − 2 = 6x + 4

In word problems, translate the sentences into an algebraic expression first, then simplify. For instance, “three times a number plus two more than the same number” becomes 3x + (x + 2) = 4x + 2.

在文字题中，先将句子转化为代数表达式，然后化简。例如，”一个数的三倍加上比这个数多二” 转化为 3x + (x + 2) = 4x + 2。

12. Exam Tips and Summary | 考试提示与总结

In the IGCSE examination, you may be asked to simplify a single expression or to simplify within a longer question. Always write your final answer with terms in a sensible order, and check that no further simplification is possible.

在 IGCSE 考试中，你可能会被要求化简一个表达式，或在一个较长的题目中进行化简。始终以合理的顺序书写最终答案，并检查是否还能继续化简。

Follow these steps in order:

按以下步骤操作：

1. Identify all like terms in the expression.

1. 识别表达式中所有的同类项。
2. Group them mentally or on paper.

2. 在脑中或在纸上对它们分组。
3. Add or subtract the coefficients of each group.

3. 对每一组的系数进行加或减。
4. Write the simplified expression, combining all groups.

4. 写出简化后的表达式，将各组组合起来。

Mastering collecting like terms will make all future algebra topics, including solving equations, factorising, and manipulating fractions, significantly easier.

掌握合并同类项将使所有后续代数主题，包括解方程、因式分解和分式运算，变得容易得多。

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Term \| 项	Type \| 类型
3x²	x² term \| x² 项
−5x and −x	x terms \| x 项
7 and −4	constant terms \| 常数项
2x²	x² term \| x² 项

Collecting Like Terms | 合并同类项