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Combining Like Terms | 合并同类项

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📚 Combining Like Terms | 合并同类项

Algebra is the language of patterns, and at the heart of this language lies the idea of simplifying expressions. Just as we group identical objects to count them more easily, we group identical variable terms to write algebraic expressions more concisely. Combining like terms is one of the most essential skills in the IGCSE Mathematics syllabus — it appears in algebraic manipulation, equation solving, factorisation, and even coordinate geometry. A solid grasp of this skill will save you time and prevent careless errors across almost every exam paper.

代数是描述规律的语言，而这一语言的核心思想之一就是化简表达式。正如我们把相同的物体分组以便更容易计数，我们也把相同的变量项分组，从而更简洁地书写代数表达式。合并同类项是 IGCSE 数学大纲中最关键的技能之一——它出现在代数运算、解方程、因式分解乃至坐标几何中。扎实掌握这一技能，将帮助你在几乎所有试卷中节省时间并避免粗心错误。

1. What Are Like Terms | 什么是同类项

A term is a single number, a single variable, or a product of numbers and variables. For example, 3x, -2y², 7ab, and 5 are all terms. The numerical factor multiplying the variable is called the coefficient. In the term 4x, the coefficient is 4; in the term -x, the coefficient is -1.

项是一个单独的数字、一个单独的变量，或者数字与变量的乘积。例如，3x、-2y²、7ab 和 5 都是项。与变量相乘的数字因数称为系数。在项 4x 中，系数是 4；在项 -x 中，系数是 -1。

Like terms are terms that have exactly the same variable part — that is, the same variables raised to the same powers. The coefficients do not need to be the same. For instance, 3x and 5x are like terms because both contain x to the power 1. Similarly, 2y² and -7y² are like terms because both contain y².

同类项是变量部分完全相同的项——也就是说，变量相同且对应指数也相同。系数不必相同。例如，3x 和 5x 是同类项，因为它们都包含一次方 x。类似地，2y² 和 -7y² 是同类项，因为它们都包含 y²。

By contrast, 3x and 3x² are not like terms because the powers of x differ. Likewise, 4x and 4y are not like terms because the variables themselves are different. Remember: both the variable and its exponent must match exactly for two terms to be like terms.

相比之下，3x 和 3x² 不是同类项，因为 x 的指数不同。同样地，4x 和 4y 也不是同类项，因为变量本身不同。请记住：只有当变量和它的指数都完全相同时，两个项才是同类项。

Expression Like Terms Present Unlike Terms
3x + 5x 3x, 5x None
2y² + 7y² 2y², 7y² None
4x + 9y None 4x, 9y
x + 2x² None x, 2x²

2. Identifying Like Terms | 识别同类项

To identify like terms, follow a simple two-step check. First, look at the variables: they must be identical. Second, look at the exponents on each variable: they must also be identical. If both checks pass, the terms are like terms; otherwise, they are unlike.

识别同类项时，只需做两步检查。第一步看变量：变量必须相同。第二步看每个变量上的指数：指数也必须相同。如果两步检查都通过，那么这两个项就是同类项；否则就不是。

Consider the expression 5ab + 3a²b – ab + 2a²b. The terms 5ab and -ab are like terms because both contain the product a × b, with each variable to the power 1. The terms 3a²b and 2a²b are also like terms because both contain a² × b. However, 5ab and 3a²b are not like terms: although they share b, the power of a differs (1 versus 2).

考虑表达式 5ab + 3a²b – ab + 2a²b。项 5ab 和 -ab 是同类项，因为它们都包含乘积 a × b，且每个变量的指数都是 1。项 3a²b 和 2a²b 也是同类项，因为它们都包含 a² × b。然而，5ab 和 3a²b 不是同类项：虽然它们都含有 b，但 a 的指数不同（1 对 2）。

It is often helpful to underline or circle groups of like terms in different colours before combining them. This visual organisation reduces mistakes, especially in long expressions. For example, in 7x – 3y + 2x + y, you might underline 7x and 2x, and circle -3y and y.

在合并之前，用不同颜色给各组同类项加下划线或画圈往往很有帮助。这种视觉化的整理能减少错误，尤其是在长表达式中。例如，在 7x – 3y + 2x + y 中，你可以给 7x 和 2x 加下划线，并给 -3y 和 y 画圈。

3. Adding and Subtracting Like Terms | 同类项的加减合并

Once you have identified like terms, you combine them by adding or subtracting their coefficients while keeping the variable part unchanged. The variable part acts as a “label” that tells you which terms belong together.

识别出同类项之后，合并的方法是对它们的系数进行加减，同时保持变量部分不变。变量部分就像一个”标签”，告诉你哪些项属于同一组。

For example, 3x + 5x = 8x. Here we add the coefficients 3 and 5, and keep the variable x. Similarly, 7y – 2y = 5y, because 7 – 2 = 5. For negative cases, -4a + 6a = 2a, because -4 + 6 = 2.

例如，3x + 5x = 8x。这里我们把系数 3 和 5 相加，并保留变量 x。类似地，7y – 2y = 5y，因为 7 – 2 = 5。对于负数的情况，-4a + 6a = 2a，因为 -4 + 6 = 2。

3x + 5x = 8x

7y – 2y = 5y

What about the term 1x? By convention, we write simply x, because multiplying by 1 changes nothing. When combining, remember that x means 1x and -x means -1x. For instance, x + x = 2x, and -x – 3x = -4x.

那么 1x 这一项呢？按照惯例，我们直接写 x，因为乘以 1 不改变任何值。在合并时，请记住 x 表示 1x，-x 表示 -1x。例如，x + x = 2x，而 -x – 3x = -4x。

x + x = 2x

-x – 3x = -4x

4. Coefficients and Constants | 系数与常数项

A constant is a term with no variable, such as 5, -3, or ½. All constants are like terms with each other, because they have no variable part at all. For example, in the expression 4x + 7 + 3x – 2, the numbers 7 and -2 are like terms, while 4x and 3x form a separate like-term group.

常数项是不含变量的项，例如 5、-3 或 ½。所有常数项之间互为同类项，因为它们根本没有变量部分。例如，在表达式 4x + 7 + 3x – 2 中，数字 7 和 -2 是同类项，而 4x 和 3x 则构成另一组同类项。

When combining constants, simply perform the ordinary arithmetic. In the example above, the constant sum is 7 – 2 = 5, and the variable sum is 4x + 3x = 7x. Therefore, the simplified expression is 7x + 5.

合并常数项时，只需做普通的算术运算。在上面的例子中，常数之和为 7 – 2 = 5，变量之和为 4x + 3x = 7x。因此，化简后的表达式为 7x + 5。

4x + 7 + 3x – 2 = 7x + 5

A common mistake is to add a constant to a variable term, such as writing 3x + 2 = 5x. This is incorrect because 3x and 2 are not like terms; they cannot be combined. The expression 3x + 2 is already in its simplest form unless more information is given.

一个常见错误是把常数项加到变量项上，例如写成 3x + 2 = 5x。这是错误的，因为 3x 和 2 不是同类项，它们不能合并。表达式 3x + 2 已经是最简形式，除非另有信息。

5. Like Terms with Powers | 含指数的同类项

When variables are raised to powers, the exponent must match exactly for terms to be like terms. The term x² is not like the term x; the term 2x² is like the term -5x² but not like the term 2x³.

当变量带有指数时，指数必须完全相同，项才是同类项。x² 与 x 不是同类项；2x² 与 -5x² 是同类项，但与 2x³ 不是同类项。

For example, combine 3x² + 5x – 2x² + x. Group the x² terms together and the x terms together:

例如，合并 3x² + 5x – 2x² + x。将 x² 项分为一组，将 x 项分为另一组：

(3x² – 2x²) + (5x + x) = x² + 6x

Notice that we cannot combine x² and x into a single term; they represent different quantities. Think of x as “one unit of x” and x² as “one unit of x times x.” Just as 3 apples plus 5 oranges cannot be written as 8 apples, 3x² plus 5x remains two separate groups.

注意，我们不能把 x² 和 x 合并成一个项；它们代表不同的量。可以把 x 想成”一个 x 单位”，把 x² 想成”一个 x 乘 x 单位”。正如 3 个苹果加 5 个橙子不能写成 8 个苹果，3x² 加 5x 也仍然是两个独立的小组。

Higher powers follow the same rule. In the expression 2a³ + a³, the terms combine to 3a³. But 2a³ + 2a² cannot be simplified further, because the exponents 3 and 2 differ.

更高的指数遵循同样的规则。在表达式 2a³ + a³ 中，两项合并为 3a³。但 2a³ + 2a² 不能再化简，因为指数 3 和 2 不同。

2a³ + a³ = 3a³

6. Simplifying Multi-Term Expressions | 多项式的化简

Expressions containing many terms can be simplified by systematically grouping all like terms. The order of terms in the final answer does not usually matter, but it is conventional to write terms in descending powers of the variable, with the constant term last.

包含许多项的表达式可以通过系统地归组所有同类项来化简。最终答案中各项的顺序通常无关紧要，但惯例是按变量的降幂排列，常数项放在最后。

Consider the expression 6x + 4 – 3x + 9 – 2x. First, identify the like terms: the x terms are 6x, -3x, and -2x; the constants are 4 and 9. Combine the x terms: 6 – 3 – 2 = 1, so we get x. Combine the constants: 4 + 9 = 13. The simplified expression is x + 13.

考虑表达式 6x + 4 – 3x + 9 – 2x。首先识别同类项：x 项是 6x、-3x 和 -2x；常数项是 4 和 9。合并 x 项：6 – 3 – 2 = 1，因此得到 x。合并常数项：4 + 9 = 13。化简后的表达式是 x + 13。

6x + 4 – 3x + 9 – 2x = x + 13

For expressions with two variables, treat each variable group separately. For example, simplify 5a + 3b – 2a + b. Combine the a terms: 5a – 2a = 3a. Combine the b terms: 3b + b = 4b. The result is 3a + 4b.

对于含两个变量的表达式，分别处理每个变量组。例如，化简 5a + 3b – 2a + b。合并 a 项：5a – 2a = 3a。合并 b 项：3b + b = 4b。结果是 3a + 4b。

5a + 3b – 2a + b = 3a + 4b

7. Combining Terms with Brackets | 含括号的合并

When an expression contains brackets, you must expand (remove) the brackets first by applying the distributive law, and only then combine like terms. The distributive law states that a(b + c) = ab + ac.

当表达式含有括号时，必须先运用分配律展开（去掉）括号，然后才能合并同类项。分配律指出 a(b + c) = ab + ac。

For example, simplify 2(x + 3) + 3(x + 1). First, expand both brackets:

例如，化简 2(x + 3) + 3(x + 1)。首先展开两个括号：

2(x + 3) = 2x + 6

3(x + 1) = 3x + 3

Now combine the like terms: 2x + 3x = 5x, and 6 + 3 = 9. The simplified expression is 5x + 9.

现在合并同类项：2x + 3x = 5x，且 6 + 3 = 9。化简后的表达式是 5x + 9。

2(x + 3) + 3(x + 1) = 5x + 9

Be especially careful with negative signs outside brackets. Simplify 4(2x – 1) – 3(x – 2). Expanding gives 8x – 4 – 3x + 6. Notice that the second bracket becomes -3x + 6, because subtracting 3(x – 2) means subtracting 3x and subtracting -6, which is adding 6. Combining like terms: 8x – 3x = 5x, and -4 + 6 = 2. The answer is 5x + 2.

要特别注意括号外的负号。化简 4(2x – 1) – 3(x – 2)。展开得到 8x – 4 – 3x + 6。注意第二个括号变成 -3x + 6，因为减去 3(x – 2) 意味着减去 3x 再减去 -6，即加上 6。合并同类项：8x – 3x = 5x，且 -4 + 6 = 2。答案为 5x + 2。

4(2x – 1) – 3(x – 2) = 5x + 2

8. Real-World Applications | 实际应用

Combining like terms is not just a classroom exercise; it appears in real-world problems involving geometry, money, and measurement. Many IGCSE questions ask you to form an expression for a perimeter or an area, then simplify it by combining like terms.

合并同类项不仅是课堂练习；它也出现在涉及几何、金钱和测量的实际问题中。许多 IGCSE 题目要求你先写出周长或面积的表达式，然后通过合并同类项进行化简。

For example, a rectangle has length (3x + 2) cm and width (2x – 1) cm. The perimeter P is given by P = 2 × length + 2 × width:

例如，一个长方形的长为 (3x + 2) cm，宽为 (2x – 1) cm。周长 P 表示为 P = 2 × 长 + 2 × 宽：

P = 2(3x + 2) + 2(2x – 1)

P = 6x + 4 + 4x – 2

P = 10x + 2

So the perimeter simplifies to (10x + 2) cm. If x = 3, the perimeter is 10 × 3 + 2 = 32 cm. This step-by-step simplification makes evaluating the expression much easier.

因此周长化简为 (10x + 2) cm。如果 x = 3，则周长为 10 × 3 + 2 = 32 cm。这种逐步化简使表达式的求值变得容易得多。

Another example: three sides of a triangle have lengths 2a + 1, a + 4, and 3a – 2. The perimeter is (2a + 1) + (a + 4) + (3a – 2) = 6a + 3. Here the a terms combine to 6a and the constants combine to 3.

另一个例子：三角形的三条边长分别为 2a + 1、a + 4 和 3a – 2。周长为 (2a + 1) + (a + 4) + (3a – 2) = 6a + 3。这里 a 项合并为 6a，常数项合并为 3。

9. Common Mistakes to Avoid | 常见错误与避坑指南

Even strong students lose marks by making careless errors when combining like terms. Here are the most common mistakes and how to avoid them.

即使是优秀的学生，也可能在合并同类项时因粗心而丢分。以下是最常见的错误以及如何避免它们。

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