Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In IGCSE Mathematics, simplifying algebraic expressions is a core skill. One of the first and most important steps is collecting like terms, which means adding or subtracting terms that have exactly the same variable part.

在 IGCSE 数学中，化简代数式是一项核心技能。其中最先也是最重要的步骤之一就是合并同类项，即将含有完全相同变量部分的项相加或相减。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that contain exactly the same variables raised to exactly the same powers. The coefficients, which are the numbers in front of the variables, can be different.

同类项是指含有完全相同的变量、且变量的指数也完全相同的项。变量前面的数字，也就是系数，可以不同。

For example, 3x and -5x are like terms because both contain x to the power of 1. Likewise, 2x² and 7x² are like terms, and 4xy and -xy are like terms.

例如，3x 和 -5x 是同类项，因为两者都含有 x 的一次方。同样，2x² 和 7x² 是同类项，4xy 和 -xy 也是同类项。

Example terms	Like or unlike?	Reason
3x and -5x	Like	Both contain x¹
2x² and 7x²	Like	Both contain x²
4xy and -xy	Like	Both contain xy
3x and 3x²	Unlike	Powers of x are different
5x and 5y	Unlike	Variables are different
2xy and 3x	Unlike	Variable combinations are different

2. Why Do We Collect Like Terms? | 为什么要合并同类项？

Collecting like terms makes an algebraic expression shorter and clearer. It is also an essential step before solving equations, factorising, substituting values, or working with formulas.

合并同类项可以让代数式更简短、更清晰。它也是解方程、因式分解、代入数值或处理公式之前必不可少的一步。

In IGCSE exams, a question may say “simplify” or “write in its simplest form”. This usually means you must collect all like terms and remove any unnecessary brackets.

在 IGCSE 考试中，题目可能会说“化简”或“写成最简形式”。这通常意味着你必须合并所有同类项，并去掉不必要的括号。

3. The Golden Rule: Add or Subtract Coefficients Only | 黄金法则：只加减系数

When collecting like terms, add or subtract the coefficients but keep the variable part exactly the same.

合并同类项时，只加减系数，变量部分保持完全不变。

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

7y – 4y = (7 – 4)y = 3y

This works because 3x and 2x both represent multiples of the same unknown amount x. Adding them gives five lots of x.

这是因为 3x 和 2x 都表示同一个未知量 x 的倍数。将它们相加，就得到 5 个 x。

4. Handling Positive and Negative Signs | 处理正号与负号

Always take the sign in front of each term as part of that term. When several terms are combined, add the coefficients using their signs.

始终把每一项前面的符号看作该项的一部分。合并多个项时，要带着符号相加系数。

For example, in 5a – 2a + 3a, the coefficients are 5, -2 and 3. Their sum is 5 – 2 + 3 = 6.

例如，在 5a – 2a + 3a 中，系数分别是 5、-2 和 3。它们的和是 5 – 2 + 3 = 6。

5a – 2a + 3a = (5 – 2 + 3)a = 6a

A negative result is possible: 4b – 7b = -3b, because 4 – 7 = -3.

结果也可能为负数：4b – 7b = -3b，因为 4 – 7 = -3。

5. Multi-Variable Terms | 含多个变量的项

For terms with more than one variable, the variable combination must match exactly. The order of the variables does not matter because multiplication is commutative, so xy is the same as yx.

对于含多个变量的项，变量的组合必须完全一致。变量顺序无关紧要，因为乘法满足交换律，所以 xy 与 yx 相同。

2xy + 5xy = 7xy

3ab – ba = 3ab – 1ab = 2ab

However, 2xy and 3x²y are not like terms because the power of x is different. Similarly, 3xy and 3x are not like terms because one has y and the other does not.

但是，2xy 和 3x²y 不是同类项，因为 x 的指数不同。同样，3xy 和 3x 也不是同类项，因为一个含有 y，另一个不含 y。

6. Powers and Indices | 幂与指数

Terms with the same base but different powers are not like terms. For example, x and x² are different because x is x¹ and x² is x squared.

底数相同但指数不同的项不是同类项。例如，x 和 x² 是不同的，因为 x 是 x¹，而 x² 是 x 的平方。

You can combine x² with x², but you cannot combine x² with x. Treat x¹ and x² as separate categories.

你可以合并 x² 与 x²，但不能合并 x² 与 x。要把 x¹ 和 x² 视为不同的类别。

2x² + 3x + 5x² – x = (2x² + 5x²) + (3x – x) = 7x² + 2x

In the answer 7x² + 2x, the two terms cannot be combined further because they are not like terms.

在答案 7x² + 2x 中，这两项不能继续合并，因为它们不是同类项。

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

When an expression contains brackets, expand the brackets first using the distributive law, then collect any like terms that appear.

当表达式中含有括号时，先用分配律去括号，然后合并出现的同类项。

For example, simplify 2(x + 3) + 3(x – 1).

例如，化简 2(x + 3) + 3(x – 1)。

2(x + 3) + 3(x – 1) = 2x + 6 + 3x – 3 = (2x + 3x) + (6 – 3) = 5x + 3

Be careful with negative signs outside brackets. In 4 – 2(x – 5), the -2 multiplies both x and -5, giving -2x + 10.

要注意括号外的负号。在 4 – 2(x – 5) 中，-2 要同时乘以 x 和 -5，得到 -2x + 10。

8. Fractions and Decimals | 分数与小数

Like terms can also have fractional or decimal coefficients. You add or subtract these coefficients using normal fraction or decimal arithmetic.

同类项也可以有分数或小数系数。你只需按照普通的分数或小数运算规则加减这些系数。

For fractions, use a common denominator before adding or subtracting.

对于分数，加减之前要先通分。

1/2 x + 1/3 x = (1/2 + 1/3)x = (3/6 + 2/6)x = 5/6 x

For decimals, align the place values and combine the numbers directly.

对于小数，对齐数位后直接合并数字。

0.5x + 0.25x = 0.75x

9. Common Mistakes to Avoid | 常见错误

Many IGCSE students lose marks by combining terms that are not actually like terms. Always check that the variable part is identical before adding or subtracting.

许多 IGCSE 学生因为合并了实际上不属于同类的项而丢分。加减之前一定要检查变量部分是否完全相同。

Mistake 1: Writing 2x + 3y as 5xy. This is wrong because x and y are different variables, so they cannot be combined.

错误一：把 2x + 3y 写成 5xy。这是错误的，因为 x 和 y 是不同的变量，不能合并。

Mistake 2: Writing 4x + 2 as 6x. This is wrong because 2 has no variable, while 4x has x.

错误二：把 4x + 2 写成 6x。这是错误的，因为 2 不含变量，而 4x 含有 x。

Mistake 3: Writing x + x² as 2x². This is wrong because x and x² have different powers.

错误三：把 x + x² 写成 2x²。这是错误的，因为 x 和 x² 的指数不同。

Mistake 4: Forgetting the sign attached to a term, for example writing 5x – 2 + 3 as 5x – 5 instead of 5x + 1.

错误四：忘记项前面的符号，例如把 5x – 2 + 3 写成 5x – 5，而不是 5x + 1。

10. Exam-Style Worked Examples | 考试型例题

Worked example 1: Simplify 7p – 2q + 3p + 5q.

例题 1：化简 7p – 2q + 3p + 5q。

7p – 2q + 3p + 5q = (7p + 3p) + (-2q + 5q) = 10p + 3q

Worked example 2: Simplify 4(2x – 3) – 2(x + 5). First expand, then collect like terms.

例题 2：化简 4(2x – 3) – 2(x + 5)。先去括号，再合并同类项。

4(2x – 3) – 2(x + 5) = 8x – 12 – 2x – 10 = (8x – 2x) + (-12 – 10) = 6x – 22

In both examples, the final answer has no like terms left and is therefore in its simplest form.

在这两个例子中，最终答案都没有剩余的同类项，因此已经是最简形式。

11. Quick Practice Questions | 快速练习

Try simplifying the following expressions before checking the answers.

在查看答案之前，试着化简以下表达式。

Practice 1: 9m + 4n – 3m + 2n

练习 1：9m + 4n – 3m + 2n

9m + 4n – 3m + 2n = (9m – 3m) + (4n + 2n) = 6m + 6n

Practice 2: 5x² + 3x – 2x² + 7x

练习 2：5x² + 3x – 2x² + 7x

5x² + 3x – 2x² + 7x = (5x² – 2x²) + (3x + 7x) = 3x² + 10x

Practice 3: 3(2a + 1) – 4(a – 2)

练习 3：3(2a + 1) – 4(a – 2)

3(2a + 1) – 4(a – 2) = 6a + 3 – 4a + 8 = (6a – 4a) + (3 + 8) = 2a + 11

12. Summary and Key Points | 总结与要点

Collecting like terms is a fundamental IGCSE algebra skill. It relies on one simple idea: only terms with exactly the same variable part can be added or subtracted.

合并同类项是 IGCSE 代数的一项基本技能。它基于一个简单的原则：只有变量部分完全相同的项才能相加或相减。

Key point 1: Identify like terms by checking the variable part, not the coefficient.

要点一：识别同类项时，要看变量部分，而不是系数。

Key point 2: Add or subtract coefficients while keeping the variable part unchanged.

要点二：加减系数，变量部分保持不变。

Key point 3: Expand brackets first when necessary, then collect like terms.

要点三：必要时先去括号，再合并同类项。

Key point 4: Never combine terms with different variables or different powers, such as x and x².

要点四：不要合并变量或指数不同的项，例如 x 和 x²。

Key point 5: Always include the correct sign of each term when adding coefficients.

要点五：加减系数时，一定要带上每一项的正确符号。

Published by TutorHao | IGCSE Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Collecting Like Terms | 合并同类项