Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, an expression is made up of terms. Simplifying an expression often means combining terms that have the same variable part. This process is called collecting like terms, and it is a key skill in IGCSE Mathematics.

在代数中，代数式由若干项组成。化简代数式通常意味着将具有相同变量部分的项合并在一起。这个过程称为“合并同类项”，是 IGCSE 数学中的关键技能。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable part, including the same letters and the same powers. The coefficients can be different. For example, 3x and -5x are like terms because both contain x to the first power.

同类项是指变量部分完全相同的项，即字母相同且对应字母的指数也相同。系数可以不同。例如，3x 和 -5x 是同类项，因为都含有一次幂的 x。

Another example: 4x² and 9x² are like terms because both contain x². However, 4x² and 4x are not like terms because the powers of x are different.

另一个例子：4x² 和 9x² 是同类项，因为都含有 x²。但 4x² 和 4x 不是同类项，因为 x 的幂次不同。

2. Coefficients and Variable Parts | 系数与变量部分

In a term like 7x³, the number 7 is called the coefficient, and x³ is called the variable part. When collecting like terms, you only change the coefficients; the variable part stays exactly the same.

在像 7x³ 这样的项中，数字 7 称为系数，x³ 称为变量部分。合并同类项时，只改变系数，变量部分保持不变。

If a term has no visible number before the variable, the coefficient is 1. For example, y means 1y, and -a means -1a.

如果一个项在变量前没有明显数字，那么系数就是 1。例如，y 表示 1y，-a 表示 -1a。

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

Collecting like terms makes algebraic expressions shorter and easier to work with. It helps you see the overall structure of an expression, and it is essential before solving equations or substituting values.

合并同类项能让代数式更简洁，也更容易处理。它帮助你看清代数式的整体结构，也是解方程或代入求值前的重要步骤。

For example, the expression 5a + 3a is easier to understand when it is written as 8a.

例如，表达式 5a + 3a 写成 8a 后更容易理解。

4. The Basic Rule: Add or Subtract Coefficients | 基本法则：对系数进行加减

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

合并同类项时，保持变量部分不变，对系数进行相加或相减。例如：

3x + 5x = (3 + 5)x = 8x

9y – 4y = (9 – 4)y = 5y

Notice that the variable x or y is written once after the coefficient. The power of the variable does not change.

注意，变量 x 或 y 在合并后的系数后面只写一次，变量的幂次不会改变。

5. Collecting More Than Two Like Terms | 合并多个同类项

An expression can contain several like terms. You can group them first, then combine the coefficients. For example:

一个代数式可能包含多个同类项。你可以先分组，再合并系数。例如：

2a + 3b – a + 4b = (2a – a) + (3b + 4b) = a + 7b

Here, a and b are different variables, so they are collected separately.

这里 a 和 b 是不同变量，因此需要分别合并。

6. Dealing with Negative Terms | 处理负项

When you see a minus sign in front of a term, treat it as a negative coefficient. For example:

当你看到某项前有减号时，应把它视为负系数。例如：

4p – 6p + p = (4 – 6 + 1)p = -1p = -p

Remember that p is the same as 1p, so the coefficient of p is 1.

请记住，p 等于 1p，因此 p 的系数是 1。

7. Terms That Cannot Be Combined | 不能合并的项

Different variables or different powers mean that terms are not like terms and cannot be combined. For example, 5x and 5y cannot be added, and 5x and 5x² cannot be combined either.

变量不同或幂次不同意味着项不是同类项，因此不能合并。例如，5x 和 5y 不能相加，5x 和 5x² 也不能合并。

The expression 3x + 2y + 4x² is already in its simplest form because no terms have the same variable part.

表达式 3x + 2y + 4x² 已经是最简形式，因为没有两项具有相同的变量部分。

8. Collecting Like Terms After Expanding Brackets | 去括号后再合并

Sometimes you need to expand brackets before collecting like terms. For example:

有时需要先去括号，再合并同类项。例如：

2(x + 3) + 4x = 2x + 6 + 4x = 6x + 6

After expanding, group the x terms together and then simplify.

展开后，把含 x 的项放在一起，再进行化简。

9. Combining Like Terms with Indices | 含幂次的同类项合并

Terms are like terms only if the variables have the same index. For example, 3x² and 5x² are like terms, but 3x² and 5x are not like terms.

只有当变量的指数相同时，两项才是同类项。例如，3x² 和 5x² 是同类项，但 3x² 和 5x 不是同类项。

When combining x² terms, add only the coefficients:

合并 x² 项时，只对系数相加：

3x² + 5x² = 8x²

Similarly, 2x²y and -7x²y are like terms because both have the same variable part x²y.

类似地，2x²y 和 -7x²y 是同类项，因为它们的变量部分都是 x²y。

10. IGCSE-Style Worked Examples | IGCSE 典型例题

Example 1: Simplify 6a + 2b – 3a + 5b.

例 1：化简 6a + 2b – 3a + 5b。

Solution: Group like terms: (6a – 3a) + (2b + 5b) = 3a + 7b.

解答：先分组：(6a – 3a) + (2b + 5b) = 3a + 7b。

Example 2: Simplify 5c + 3c² – 2c + c².

例 2：化简 5c + 3c² – 2c + c²。

Solution: (5c – 2c) + (3c² + c²) = 3c + 4c².

解答：(5c – 2c) + (3c² + c²) = 3c + 4c²。

Example 3: Simplify 2(3x + 1) – 4x.

例 3：化简 2(3x + 1) – 4x。

Solution: Expand first: 6x + 2 – 4x = 2x + 2.

解答：先去括号：6x + 2 – 4x = 2x + 2。

11. Common Mistakes to Avoid | 常见易错点

Do not combine terms with different variables. For example, 3x + 4y cannot be simplified to 7xy.

不要把不同变量的项合并。例如，3x + 4y 不能化简为 7xy。
Do not change the exponent when combining. For example, 3x + 4x = 7x, not 7x².

合并时不要改变指数。例如，3x + 4x = 7x，而不是 7x²。
Be careful with signs. For example, -2x + 5x = 3x, not -3x.

注意正负号。例如，-2x + 5x = 3x，而不是 -3x。
Do not forget the coefficient 1. For example, x + 2x = 3x, and x is 1x.

不要忘记系数 1。例如，x + 2x = 3x，其中 x 是 1x。

12. Summary | 总结

Collecting like terms is the process of simplifying expressions by adding or subtracting the coefficients of terms that have identical variable parts. Only the coefficients change; the variable part remains unchanged. This skill is used throughout IGCSE Mathematics, from algebra to solving equations.

合并同类项是指通过将具有相同变量部分的项的系数相加或相减来化简代数式的过程。只有系数会改变，变量部分保持不变。这一技能在 IGCSE 数学中广泛使用，从代数运算到解方程都会用到。

Practice grouping like terms carefully, check signs, and remember the variables must match exactly before you combine.

练习时注意分组、核对正负号，并牢记只有在变量完全匹配时才能合并。

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Collecting Like Terms | 合并同类项