Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, expressions often contain multiple terms that look similar but are not identical. Learning to combine like terms is one of the most important foundational skills for IGCSE Mathematics, because it allows us to simplify expressions, solve equations, and tackle more advanced topics with confidence.

在代数中，表达式常常包含多个看起来相似但并不完全相同的项。学会合并同类项是IGCSE数学中最基础也最重要的技能之一，因为它能帮助我们化简表达式、解方程，并自信地处理更高级的课题。

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1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part — the same letters raised to the same powers. The numerical coefficients (the numbers in front of the variables) can be different, but the variable part must match exactly.

同类项是指变量部分完全相同的项——即字母相同且字母的指数也相同。变量前面的数字系数（即系数）可以不同，但变量部分必须完全一致。

For example:

• 3x and 5x are like terms, because both contain the single variable x¹.
• 3x 和 5x 是同类项，因为两者都含有变量 x¹。
• 4x² and −2x² are like terms, because both contain x².
• 4x² 和 −2x² 是同类项，因为两者都含有 x²。
• 7xy and 3yx are like terms, because xy = yx (commutative property of multiplication).
• 7xy 和 3yx 是同类项，因为 xy = yx（乘法交换律）。

By contrast, 3x and 3x² are not like terms, because the powers of x are different. Similarly, 2xy and 2x are not like terms, because the variable parts differ.

相比之下，3x 和 3x² 不是同类项，因为 x 的指数不同。同理，2xy 和 2x 也不是同类项，因为变量部分不同。

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2. Identifying Like Terms | 识别同类项

To identify like terms, look only at the variable part of each term. Ignore the coefficient when deciding whether two terms are "like". The coefficient only matters when we actually combine the terms.

在识别同类项时，只需要看每一项的变量部分。判断两项是否为"同类"时，忽略系数；系数只在我们真正合并时才起作用。

Consider the expression:

5a + 3b − 2a + 7b + 4

Here, 5a and −2a are like terms (variable a), and 3b and 7b are like terms (variable b). The number 4 is a constant term, and constants are also like terms with each other.

这里，5a 和 −2a 是同类项（变量都是 a），3b 和 7b 是同类项（变量都是 b）。数字 4 是常数项，而常数项之间也是同类项。

A helpful strategy is to underline or circle different families of like terms with different symbols before combining them. This reduces the chance of missing a term during simplification.

一个有用的策略是：在合并之前，用不同的符号（如下划线或圆圈）标记不同组的同类项。这样可以减少在化简过程中漏掉某项的机会。

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3. Combining Terms with Addition and Subtraction | 用加减法合并项

When combining like terms, add or subtract the coefficients and keep the variable part unchanged. This works because of the distributive property:

合并同类项时，将系数相加或相减，变量部分保持不变。这是依据乘法分配律实现的：

ax + bx = (a + b)x

For example:

• 3x + 5x = (3 + 5)x = 8x
• 3x + 5x = (3 + 5)x = 8x
• 7y − 2y = (7 − 2)y = 5y
• 7y − 2y = (7 − 2)y = 5y
• −4p + 6p = (−4 + 6)p = 2p
• −4p + 6p = (−4 + 6)p = 2p
• 9m − 9m = 0m = 0
• 9m − 9m = 0m = 0

Note that when the coefficient is 1, we usually do not write it. For example, 1x is written simply as x. When combining terms such as x + x, the result is 2x.

注意：当系数为 1 时，我们通常不写出来。例如，1x 直接写成 x。合并如 x + x 这样的项时，结果为 2x。

m. For example, in 6a − 3a, the second term is −3a, so we compute 6 + (−3) = 3, giving 3a.

还要记住：每一项前面的符号属于该项。例如，在 6a − 3a 中，第二项是 −3a，所以我们计算 6 + (−3) = 3，得到 3a。

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4. Constants Are Like Terms Too | 常数也是同类项

Constant terms — numbers without variables — are considered like terms with each other. You can combine them just as you combine variable terms, by adding or subtracting their values.

常数项——即不含变量的数字——彼此之间也是同类项。你可以像合并变量项一样合并它们，只需将数值相加或相减。

For example:

8 + 3 − 5 = 6

able terms first (4x + 2x = 6x) and then combine the constants (8 − 5 = 3), giving 6x + 3.

因此，在表达式 4x + 8 + 2x − 5 中，我们先合并变量项（4x + 2x = 6x），再合并常数项（8 − 5 = 3），最终得到 6x + 3。

Be careful not to combine constants with variable terms. The expression 6x + 3 cannot be simplified further because 6x and 3 are not like terms.

注意不要将常数项与变量项合并。表达式 6x + 3 无法再化简，因为 6x 和 3 不是同类项。

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5. Simplifying Multi-Variable Expressions | 化简多变元表达式

When an expression contains more than one variable, group each family of like terms separately. The order in which you write the final answer does not matter, but it is common to write terms in alphabetical order or with the highest power first.

当表达式中包含多个变量时，将每一组同类项分别归类合并。最终答案的书写顺序不固定，但通常按字母顺序排列，或按指数从高到低排列。

Worked example — simplify:

5x + 7y − 3x + 2y + 4x

Step 1: Identify like families: x-terms are 5x, −3x, 4x; y-terms are 7y, 2y.

第一步：识别同类组——含 x 的项：5x、−3x、4x；含 y 的项：7y、2y。

Step 2: Combine x-terms: 5 − 3 + 4 = 6, so we have 6x.

第二步：合并含 x 的项：5 − 3 + 4 = 6，得到 6x。

Step 3: Combine y-terms: 7 + 2 = 9, so we have 9y.

第三步：合并含 y 的项：7 + 2 = 9，得到 9y。

Step 4: Write the simplified expression: 6x + 9y.

第四步：写出化简后的表达式：6x + 9y。

Notice that 6x and 9y remain separate because they are not like terms. The answer is 6x + 9y, not 15xy.

请注意，6x 和 9y 不能合并，因为它们不是同类项。答案是 6x + 9y，而不是 15xy。

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6. Step-by-Step Worked Example | 一步一步的完整示例

Let us work through a more complex example to show the full process.

让我们通过一个更复杂的例子来展示完整的过程。

Simplify:

3x² + 5x − 2 + 4x² − 3x + 7

Step 1: Group the like terms by variable and power.

第一步：按变量和指数对同类项分组。

• x² terms: 3x² and 4x²
• x² 项：3x² 和 4x²
• x terms: 5x and −3x
• x 项：5x 和 −3x
• Constants: −2 and 7
• 常数项：−2 和 7

Step 2: Combine each group separately.

第二步：分别合并每一组。

3x² + 4x² = 7x²

5x − 3x = 2x

−2 + 7 = 5

Step 3: Write the final answer:

第三步：写出最终答案：

7x² + 2x + 5

Always check whether any further simplification is possible. Here, 7x², 2x and 5 are all unlike, so the expression is fully simplified.

始终检查是否还能进一步化简。这里 7x²、2x 和 5 都互不相同，因此表达式已完全化简。

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7. Distribution and Combining | 先去括号再合并

In IGCSE exams, expressions often include brackets. Before combining like terms, you must first expand any brackets using the distributive property.

在IGCSE考试中，表达式常常包含括号。在合并同类项之前，必须先利用分配律展开括号。

Recall the distributive property:

a(b + c) = ab + ac

For example, simplify 3(2x + 4) + 5x:

例如，化简 3(2x + 4) + 5x：

Step 1: Expand: 3(2x + 4) = 6x + 12.

第一步：展开：3(2x + 4) = 6x + 12。

Step 2: The expression becomes 6x + 12 + 5x.

第二步：表达式变为 6x + 12 + 5x。

Step 3: Combine like terms: 6x + 5x = 11x, so the answer is 11x + 12.

第三步：合并同类项：6x + 5x = 11x，答案为 11x + 12。

10. The sign of every term inside the bracket changes when the number outside is negative.

特别小心括号外的负号。例如，4x − 2(3x − 5) 表示 4x − 2 × 3x − 2 × (−5) = 4x − 6x + 10 = −2x + 10。当括号外的数为负数时，括号内每一项的符号都会改变。

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8. Common Mistakes to Avoid | 需要避免的常见错误

Many students lose marks in IGCSE algebra questions by making simple mistakes when combining like terms. Here are the most common ones to watch out for.

许多学生在IGCSE代数题中因合并同类项时的简单错误而失分。以下是需要特别留意的常见错误。

• Mistake 1: Combining unlike terms. Writing 3x + 4y = 7xy is wrong. Unlike terms cannot be combined.
• 错误一：合并非同类项。把 3x + 4y 写成 7xy 是错误的。非同类项不能合并。
• Mistake 2: Changing the variable part. x + x is 2x, not x². Only coefficients are added; the variable part stays the same.
• 错误二：改变变量部分。x + x 等于 2x，而不是 x²。只有系数相加，变量部分保持不变。
• Mistake 3: Forgetting the sign. 5 − 2 + 3)a = 6a. But in 5a − (2a + 3a), the brackets change everything: 5a − 2a − 3a = 0.
• 错误三：忽略符号。在 5a − 2a + 3a 中，合并得到 (5 − 2 + 3)a = 6a。但在 5a − (2a + 3a) 中，括号改变了一切：5a − 2a − 3a = 0。
• Mistake 4: Misidentifying x² and x. 2x² + 3x cannot be simplified to 5x³ or 5x. The powers must match.
• 错误四：混淆 x² 与 x。2x² + 3x 不能化简为 5x³ 或 5x。指数必须相同才能合并。

A good habit is to always re-read the original expression after simplifying and verify that no terms were missed or altered.

一个好习惯是：化简完成后重新阅读原表达式，确认没有遗漏或改变任何项。

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9. Exam-Style Questions | 考试题型练习

Here are some typical IGCSE-style questions on combining like terms. Try them yourself before reading the solutions.

以下是一些典型的IGCSE合并同类项题型。请先自己尝试，再看解答。

Question 1: Simplify 6x + 3y − 2x + 5y.

题目1：化简 6x + 3y − 2x + 5y。

Solution: x-terms: 6x − 2x = 4x. y-terms: 3y + 5y = 8y. Answer: 4x + 8y.

解答：含 x 项：6x − 2x = 4x；含 y 项：3y + 5y = 8y。答案：4x + 8y。

Question 2: Simplify 5a² + 2ab − 3a² + 7ab − b.

题目2：化简 5a² + 2ab − 3a² + 7ab − b。

a² = 2a². ab terms: 2ab + 7ab = 9ab. The term −b has no like terms. Answer: 2a² + 9ab − b.

解答：a² 项：5a² − 3a² = 2a²；ab 项：2ab + 7ab = 9ab；−b 项没有同类项。答案：2a² + 9ab − b。

Question 3: Simplify 2(3x − 4) − (x + 5).

题目3：化简 2(3x − 4) − (x + 5)。

coefficient of −1: −(x + 5) = −x − 5. Combine: 6x − x = 5x; −8 − 5 = −13. Answer: 5x − 13.

解答：展开：2(3x − 4) = 6x − 8。第二个括号前有隐含系数 −1：−(x + 5) = −x − 5。合并：6x − x = 5x；−8 − 5 = −13。答案：5x − 13。

Question 4: Simplify x² + 3x − 2x² + 4.

题目4：化简 x² + 3x − 2x² + 4。

Solution: x² terms: x² − 2x² = −x². The 3x and 4 have no like terms. Answer: −x² + 3x + 4.

解答：x² 项：x² − 2x² = −x²；3x 和 4 没有同类项。答案：−x² + 3x + 4。

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10. Summary and Key Points | 总结与要点

Combining like terms is a core skill that appears in nearly every IGCSE Mathematics paper. Mastering it will make solving equations, factorising and working with algebraic fractions much easier.

合并同类项是几乎每张IGCSE数学试卷都会考查的核心技能。掌握它，将使解方程、因式分解以及处理代数分式变得容易得多。

Key points to remember:

需要记住的要点：

• Like terms have identical variable parts, including the same powers.
• 同类项具有完全相同的变量部分，包括相同的指数。
• Combine only the coefficients; keep the variable part unchanged.
• 只合并系数；变量部分保持不变。
• Constants are like terms with each other but not with variable terms.
• 常数项之间互为同类项，但不能与变量项合并。
• Always expand brackets before combining like terms.
• 合并同类项之前，务必先展开括号。
• Check your final answer to ensure no term has been missed.
• 检查最终答案，确保没有遗漏任何项。

With regular practice, combining like terms will become automatic. Keep a clear, organised layout in your working, and you will consistently earn full marks on these questions.

通过定期练习，合并同类项将变得得心应手。在解题过程中保持清晰、有条理的书写，你就能在这些题目上稳定地获得满分。

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