Simplifying Algebraic Expressions: Combining Like Terms | 合并同类项：代数式化简

📚 Simplifying Algebraic Expressions: Combining Like Terms | 合并同类项：代数式化简

In algebra, simplifying an expression means rewriting it in its most compact and manageable form without changing its value. This is a core skill covered in IGCSE Mathematics, and it relies on the operation known as “combining like terms.”

在代数中，化简表达式意味着在不改变其数值的前提下，将其改写为最紧凑、最易于处理的形式。这是 IGCSE 数学的核心技能之一，而这一技能的关键操作就是”合并同类项”。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable parts raised to the same powers. The coefficients (the numbers in front of the variables) may be different, but the variables and their exponents must match exactly.

同类项是指变量部分及其指数完全相同的项。系数（变量前面的数字）可以不同，但变量及其指数必须完全一致。

For example, 3x and 5x are like terms because both have the variable x to the power of 1. Similarly, 4x² and −2x² are like terms because both have the variable x squared.

例如，3x 和 5x 是同类项，因为它们的变量都是 x 的 1 次方。同样，4x² 和 −2x² 也是同类项，因为它们的变量都是 x 的平方。

However, 2x and 2x² are NOT like terms, because the exponents differ. Similarly, xy and x²y are not like terms, since the power of x differs between them.

然而，2x 和 2x² 不是同类项，因为指数不同。同样，xy 和 x²y 也不是同类项，因为其中 x 的幂次不同。

Like terms: same variable(s), same exponent(s) | 同类项：相同的变量，相同的指数

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

Combining like terms simplifies an expression so we can evaluate it more easily, substitute values more quickly, and solve equations more efficiently. It reduces the overall number of terms and makes the structure of the expression clearer.

合并同类项能够化简表达式，使我们更容易代入数值求值、更快地进行变量替换，并更高效地求解方程。它减少了项的总数，使表达式的结构更加清晰。

Consider a real-life scenario: if you buy 3 apples for 2 dollars each and then 5 more apples for 2 dollars each, instead of calculating 3×2+5×2 separately, you can combine the apples: (3+5)×2=16 dollars. Algebra follows the same intuitive logic.

考虑一个生活场景：如果你以每个 2 美元的价格买了 3 个苹果，然后又买了 5 个，与其分别计算 3×2+5×2，不如先把苹果合并：(3+5)×2=16 美元。代数遵循同样的直观逻辑。

It is also an essential prerequisite for solving linear equations, expanding brackets, factorising, and working with algebraic fractions.

它也是求解线性方程、去括号、因式分解和处理代数分式的重要前提。

3. Identifying Terms and Coefficients | 识别项与系数

Before combining, we must identify each term in the expression. A term is a product of a coefficient and a variable part, separated by + or − signs. The coefficient is the numerical factor, and the variable part contains letters and their powers.

在合并之前,我们必须先识别表达式中的每一项。项是系数与变量部分的乘积，各单项之间以加号或减号分隔。系数是数字因子，变量部分则包含字母及其幂次。

In the expression 7x + 3 − 2x + 5, the terms are 7x, 3, −2x, and 5. The terms 7x and −2x are like terms, while 3 and 5 are constant terms and are also like terms with each other.

在表达式 7x + 3 − 2x + 5 中，各项分别是 7x、3、−2x 和 5。其中 7x 与 −2x 是同类项，而 3 和 5 是常数项，它们彼此也是同类项。

Constant terms are terms with no variables, such as 4 or −9. They can always be combined with each other.

常数项是不含变量的项，例如 4 或 −9。它们总是可以互相合并。

Term = coefficient × variable part | 项 = 系数 × 变量部分
+ and − signs belong to the term that follows them | 加号和减号属于其后的那个项

4. The Basic Rule for Combining Like Terms | 合并同类项的基本法则

The basic rule is simple: add or subtract the coefficients of the like terms, and keep the variable part exactly the same. The variable part does not change during the operation.

基本法则很简单：对同类项的系数进行相加或相减，变量部分保持不变。变量部分在运算过程中不发生变化。

ax + bx = (a + b)x | ax − bx = (a − b)x

For example, 4x + 3x = (4+3)x = 7x. Similarly, 9y − 5y = (9−5)y = 4y.

例如，4x + 3x = (4+3)x = 7x。同样，9y − 5y = (9−5)y = 4y。

Remember that when a term has no visible coefficient, like x, the coefficient is actually 1. So x + 2x = (1+2)x = 3x.

注意，如果某项没有写出系数，比如 x，实际上它的系数是 1。所以 x + 2x = (1+2)x = 3x。

5. Combining Terms with Identical Squared Terms | 合并含平方的同类项

In IGCSE Mathematics, you will often encounter terms with powers. The rule is unchanged: only combine terms where the variable part is identical, including the exponent.

在 IGCSE 数学中，你经常会遇到含有幂的项。法则不变：只有变量部分（包括指数）完全相同的项才能合并。

For the expression 3x² + 5x², we combine: (3+5)x² = 8x². The x² part is preserved exactly.

对于表达式 3x² + 5x²，我们可以合并为：(3+5)x² = 8x²。其中的 x² 部分被原样保留。

Consider a longer example: 2x² + 3x + 4x² + x. Group the like terms: the x² terms are 2x² and 4x²; the x terms are 3x and x. Combining gives 6x² + 4x.

来看一个较长的例子：2x² + 3x + 4x² + x。先将同类项分组：含 x² 的项是 2x² 和 4x²；含 x 的项是 3x 和 x。合并后得到 6x² + 4x。

It is critical not to combine 3x with 4x². They are not like terms because their exponents are different, and combining them would produce an incorrect result.

需要注意，不能将 3x 和 4x² 合并。它们不是同类项，因为指数不同，强行合并会产生错误的结果。

6. Combining Terms with Multiple Variables | 合并含多个变量的同类项

When terms contain two or more variables, both the variables and their exponents must match for the terms to be like terms. For example, 2xy and 7xy are like terms, because the product xy is identical in both.

当项中含有两个或多个变量时，只有变量及其指数全部匹配，这些项才是同类项。例如，2xy 和 7xy 是同类项，因为两者的乘积 xy 完全相同。

But 2xy and 2x²y are NOT like terms, because in the first term x has exponent 1, while in the second term x has exponent 2. Similarly, 3ab and 3a²b are not like terms.

但 2xy 和 2x²y 不是同类项，因为第一项中 x 的指数是 1，而第二项中 x 的指数是 2。同理，3ab 和 3a²b 也不是同类项。

Example: Simplify 4ab + 5a − 3ab + 2a. The ab terms are 4ab and −3ab, giving ab. The a terms are 5a and 2a, giving 7a. The simplified expression is ab + 7a.

例如：化简 4ab + 5a − 3ab + 2a。含 ab 的项是 4ab 和 −3ab，合并为 ab。含 a 的项是 5a 和 2a，合并为 7a。化简后的表达式为 ab + 7a。

Some expressions in IGCSE also contain mixed terms such as xy and yx. Since multiplication is commutative, xy and yx are the same, and they can be combined.

在 IGCSE 中，有些表达式还包含 xy 和 yx 这类混合项。由于乘法满足交换律，xy 和 yx 是相同的，可以合并。

7. Simplifying Expressions with Brackets | 化简含括号的表达式

When an expression contains brackets, we use the distributive law to remove the brackets first, then combine like terms. This is written as a(b + c) = ab + ac.

当表达式含有括号时，我们先用乘法分配律去掉括号，然后再合并同类项。即 a(b + c) = ab + ac。

Consider 3(x + 4) + 2(x − 1). First expand each bracket: 3x + 12 + 2x − 2. Then group like terms: 3x + 2x = 5x, and 12 − 2 = 10. The final expression is 5x + 10.

考虑 3(x + 4) + 2(x − 1)。先展开每个括号：3x + 12 + 2x − 2。然后合并同类项：3x + 2x = 5x，12 − 2 = 10。最终表达式为 5x + 10。

When subtracting a bracket, be careful with the negative sign: the sign of every term inside the bracket must change. For example, 4x − (2x + 3) = 4x − 2x − 3 = 2x − 3.

减去一个括号时要格外小心负号：括号内每一项的符号都必须改变。例如，4x − (2x + 3) = 4x − 2x − 3 = 2x − 3。

This step is often the main source of sign errors in IGCSE exams. Always rewrite the subtraction as “adding the opposite” if it helps.

这一步通常是 IGCSE 考试中符号错误的主要来源。如有必要，可以把减法改写为”加上相反数”来避免出错。

8. Handling Fractional Coefficients | 处理分数系数

Like terms with fractional coefficients follow exactly the same rules. You only need to apply the usual rules for adding and subtracting fractions to the coefficients.

含分数系数的同类项遵循完全相同的规则。你只需要对系数施以通常的分数加减运算法则即可。

Example: (1/2)x + (1/3)x. To combine, add the fractions: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, so the result is (5/6)x.

例如：(1/2)x + (1/3)x。合并时，先将分数相加：1/2 + 1/3 = 3/6 + 2/6 = 5/6，因此结果为 (5/6)x。

For subtraction, (3/4)y − (1/2)y = (3/4 − 2/4)y = (1/4)y.

对于减法，(3/4)y − (1/2)y = (3/4 − 2/4)y = (1/4)y。

Fractional coefficients appear frequently in IGCSE papers, especially in problems involving geometry formulas or word problems. Practising with fractions is essential for fluency.

分数系数在 IGCSE 考试中很常见，尤其是在涉及几何公式或应用题的题目中。熟练处理分数至关重要。

9. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Many students make predictable mistakes when combining like terms. Recognising these errors is the first step to avoiding them.

许多学生在合并同类项时会犯一些可预见的错误。识别这些错误是避免它们的第一步。

Mistake 1: Combining unlike terms. Forgetting that x and x² cannot be added. Avoid this by always checking the exponents before combining.
错误一：合并非同类的项。忘记 x 与 x² 不能相加。合并前务必检查指数以避免此错误。
Mistake 2: Incorrectly handling negative signs. For example, simplifying 5x − 3x + 2 as 2x + 2 is fine, but many students write 2x + 2 as the result of 5x − (3x + 2), which is wrong. Always distribute negative signs properly.
错误二：负号处理错误。例如，5x − (3x + 2) 的正确结果是 2x − 2，而不是 2x + 2。务必正确分配负号。

Mistake 3: Losing the coefficient 1. Writing x + 2x = 2x instead of 3x. Remember that x has an implied coefficient of 1.
错误三：丢掉系数 1。把 x + 2x 错写成 2x 而不是 3x。记住 x 隐含系数为 1。
Mistake 4: Mixing up constant terms. Constants can only combine with constants. 5 and 5x cannot be added.
错误四：混淆常数项。常数只能与常数合并。5 和 5x 不能相加。

To prevent these mistakes, write out each step clearly, circle like terms with the same colour, and always verify your final answer by substituting a simple value such as x = 1.

为预防这些错误，建议每一步都写清楚，用相同的标记圈出同类项，并通过代入一个简单数值（如 x = 1）来验证最终答案。

10. Step-by-Step Problem Solving Strategy | 分步解题策略

When facing any simplification problem in an IGCSE exam, follow a systematic approach to minimise errors and save time.

在 IGCSE 考试中面对任何化简题时，都应遵循系统化的步骤，以减少错误并节省时间。

Step 1: Scan the expression and identify all distinct variable parts and constants.

第一步：快速浏览表达式，识别所有不同的变量部分和常数。

Step 2: Rearrange if necessary, grouping like terms together. Keep the sign with each term when moving it.

第二步：如有必要，可重新排列，将同类项放在一起。移动项时务必连同其符号一起移动。

Step 3: Combine the coefficients of each group. Preserve the variable part exactly.

第三步：分别合并每组项的系数。变量部分保持不变。

Step 4: Write the simplified expression in a standard order: usually constants last, then terms in descending powers of the variable.

第四步：按标准顺序书写化简后的表达式：通常常数放在最后，变量的幂次按降序排列。

Applying this strategy, the expression 7x + 3 − 2x + 5 simplifies to 5x + 8 in four clear steps.

应用这一策略，表达式 7x + 3 − 2x + 5 可以通过清晰的四个步骤化简为 5x + 8。

11. Worked Examples | 典型例题

Let us work through several examples that represent the types of questions typically seen in IGCSE examinations.

让我们一起来做几道代表 IGCSE 考试中常见题型的例题。

Example 1: Simplify 6a + 3b − 2a + 4b.

例 1：化简 6a + 3b − 2a + 4b。

Group the a terms: 6a − 2a = 4a. Group the b terms: 3b + 4b = 7b. Answer: 4a + 7b.

将含 a 的项合并：6a − 2a = 4a。将含 b 的项合并：3b + 4b = 7b。答案为 4a + 7b。

Example 2: Simplify 2(x + 3) − (x − 5).

例 2：化简 2(x + 3) − (x − 5)。

Expand: 2x + 6 − x + 5. Combine: 2x − x = x, and 6 + 5 = 11. Answer: x + 11.

展开：2x + 6 − x + 5。合并：2x − x = x，6 + 5 = 11。答案为 x + 11。

Example 3: Simplify 5x² + 3x − 2x² − x + 7.

例 3：化简 5x² + 3x − 2x² − x + 7。

Group the x² terms: 5x² − 2x² = 3x². Group the x terms: 3x − x = 2x. The constant is 7. Answer: 3x² + 2x + 7.

合并含 x² 的项：5x² − 2x² = 3x²。合并含 x 的项：3x − x = 2x。常数为 7。答案为 3x² + 2x + 7。

Example 4: Simplify (1/2)p + (1/4)q − (1/4)p + (3/4)q.

例 4：化简 (1/2)p + (1/4)q − (1/4)p + (3/4)q。

Combine the p terms: (1/2)p − (1/4)p = (2/4 − 1/4)p = (1/4)p. Combine the q terms: (1/4)q + (3/4)q = (4/4)q = q. Answer: (1/4)p + q.

合并含 p 的项：(1/2)p − (1/4)p = (2/4 − 1/4)p = (1/4)p。合并含 q 的项：(1/4)q + (3/4)q = (4/4)q = q。答案为 (1/4)p + q。

12. Practice and Final Summary | 练习与总结

Try this quick set of practice problems to test your understanding before your next exam.

在下次考试之前，试试下面这组快速练习，检验你的理解程度。

Simplify 8x + 3x − 5x. | 化简 8x + 3x − 5x。
Simplify 4y² + 3y − 2y² + y. | 化简 4y² + 3y − 2y² + y。
Simplify 3(m + 2) − 2(m − 1). | 化简 3(m + 2) − 2(m − 1)。
Simplify (2/3)a + (1/6)a. | 化简 (2/3)a + (1/6)a。

Answers: 6x; 2y² + 4y; m + 8; (5/6)a. If you got all of these correct, you have mastered the essentials of combining like terms!

答案：6x；2y² + 4y；m + 8；(5/6)a。如果全部答对，说明你已经掌握了合并同类项的核心要点！

Combining like terms is a fundamental skill in algebra that appears consistently throughout IGCSE Mathematics. Master it early, and you will build a strong foundation for all future algebra topics.

合并同类项是代数中的一项基本技能，在 IGCSE 数学中反复出现。尽早掌握它，你将为此后所有代数主题打下坚实的基础。

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Simplifying Algebraic Expressions: Combining Like Terms | 合并同类项：代数式化简