Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, expressions often contain several terms that can be simplified by adding or subtracting them. This process is called combining like terms. It is one of the most essential skills for solving equations and simplifying algebraic expressions.

在代数中，表达式通常包含多个可以通过加减法进行简化的项。这个过程被称为合并同类项，也是解方程和化简代数表达式最基本、最重要的技能之一。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable part, meaning the same variables raised to the same powers. Constant terms (numbers without variables) are also like terms with each other.

同类项是指变量部分完全相同的项，即相同的字母且字母的指数也相同。常数项（不含字母的数）彼此之间也是同类项。

3x and 7x are like terms.
3x 和 7x 是同类项。
-2y² and 5y² are like terms.
-2y² 和 5y² 是同类项。
4 and -9 are like terms.
4 和 -9 是同类项。
3x and 3x² are not like terms.
3x 和 3x² 不是同类项。

2. Why Combine Like Terms? | 为什么要合并同类项

Combining like terms makes algebraic expressions shorter and easier to work with. It also reduces the risk of mistakes when solving equations and is often required before you can factorise or evaluate an expression.

合并同类项能让代数表达式变得更简短、更容易处理。它还能降低解方程时的出错率，而且在进行因式分解或代入求值之前，往往必须先合并同类项。

It simplifies expressions before solving equations.
它在解方程之前将表达式简化。
It helps you see the structure of the expression clearly.
它帮助你更清晰地看清表达式的结构。
It is a prerequisite for many topics such as factorisation and fractions.
它是因式分解、分式运算等许多专题的前提。

3. How to Identify Like Terms | 如何识别同类项

To decide whether two terms are like terms, ignore the number in front of the variable (the coefficient). Compare only the variable letters and their exponents. If the variables and exponents match exactly, the terms are like terms.

要判断两个项是否为同类项，只需忽略变量前面的数（即系数），只比较变量字母及其指数。如果变量和指数完全一致，这两个项就是同类项。

Same variable letters? Check if all letters are identical.
变量字母是否相同？检查所有字母是否完全一致。
Same exponents? Check that each letter has the same power.
指数是否相同？检查每个字母对应的次方是否一致。
Do not check the coefficient when determining likeness.
判断同类项时不要看系数。

4. The Process of Combining | 合并同类项的过程

Once you have identified like terms, combine them by adding or subtracting their coefficients. The variable part remains exactly the same.

一旦识别出同类项，就可以通过加或减它们的系数来合并它们。变量部分保持完全不变。

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

In the example above, the coefficient 3 and 5 are added together, while the variable x stays unchanged.

在上面的例子中，系数 3 和 5 相加，而变量 x 保持不变。

Step 1: Group the like terms together.
第一步：把同类项放在一起。
Step 2: Add or subtract the coefficients.
第二步：对系数进行加或减。
Step 3: Write the result with the variable part unchanged.
第三步：写出结果，变量部分保持不变。

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

When combining like terms, pay careful attention to the sign in front of each term. The sign belongs to the coefficient of that term.

合并同类项时，要特别留意每个项前面的正负号。这个符号属于该项的系数。

7a – 3a + 2a = (7 – 3 + 2)a = 6a

Notice that the negative sign before 3a is used as part of the coefficient -3. A common method is to treat subtraction as adding a negative number.

注意 3a 前面的负号作为系数 -3 的一部分参与运算。一种常见的方法是把减法看作加上一个负数。

Example: 5m – 8m = (5 – 8)m = -3m.
例如：5m – 8m = (5 – 8)m = -3m。
Example: -2p + 9p – 4p = (-2 + 9 – 4)p = 3p.
例如：-2p + 9p – 4p = (-2 + 9 – 4)p = 3p。

6. Combining Terms with Different Variables | 合并所含变量不同的项

Terms with different variables or different exponents cannot be combined. They must remain as separate terms in the final expression.

所含变量不同或指数不同的项不能合并，它们在最终表达式中必须保留为独立的项。

2x + 3y

This expression cannot be simplified further because x and y are different variables.

这个表达式不能再简化，因为 x 和 y 是不同的变量。

4x² + 2x – x² + 5x = (4x² – x²) + (2x + 5x) = 3x² + 7x

Here x² and x are not like terms, so they are combined separately.

这里 x² 和 x 不是同类项，因此分别合并。

7. Full Worked Examples | 完整例题示范

Let us apply the rules to a more complicated expression. Simplify: 5x + 3y – 2x + y.

我们把规则应用到更复杂的表达式上。化简：5x + 3y – 2x + y。

Group the x terms: 5x – 2x.
将含 x 的项分组：5x – 2x。
Group the y terms: 3y + y.
将含 y 的项分组：3y + y。
Combine each group: 5x – 2x = 3x, and 3y + y = 4y.
分别合并：5x – 2x = 3x，3y + y = 4y。

5x + 3y – 2x + y = 3x + 4y

Another example: Simplify 6a² – 4a + 2a² + 7a – 3.

另一个例子：化简 6a² – 4a + 2a² + 7a – 3。

(6a² + 2a²) + (-4a + 7a) – 3 = 8a² + 3a – 3

Notice that the constant term -3 has no like term here, so it remains unchanged.

注意常数项 -3 在这里没有同类项，因此保持不变。

8. Common Mistakes | 常见错误

Many students make the same errors when combining like terms. Being aware of these mistakes can help you avoid them in the exam.

很多学生在合并同类项时会出现同样的错误。了解这些错误有助于你在考试中避免它们。

Combining unlike terms: 2x + 3y = 5xy is incorrect.
合并非同类项：2x + 3y = 5xy 是错误的。
Mixing up powers: 2x + 3x² = 5x³ is incorrect.
混淆指数：2x + 3x² = 5x³ 是错误的。
Losing negative signs: 7a – 5a = 2a is correct, but 7a – 5a = 12a is wrong.
漏掉负号：7a – 5a = 2a 是正确的，而 7a – 5a = 12a 是错误的。
Forgetting to keep the variable part unchanged.
忘记保持变量部分不变。

9. Exam Tips | 考试技巧

In the IGCSE mathematics exam, you will often be asked to simplify expressions before solving equations. These tips will help you score full marks.

在 IGCSE 数学考试中，你常常需要先化简表达式再解方程。以下技巧能帮助你获得满分。

Use a pencil to underline all like terms in the same way.
用铅笔以相同的记号把所有同类项标记出来。
Rearrange the expression so that like terms are next to each other.
重新排列表达式，使同类项相邻。
Always show your grouping steps clearly to earn method marks.
清晰写出分组步骤，以获得方法分。
After simplifying, quickly check by substituting a simple number, such as 1.
化简后，代入一个简单的数（如 1）快速检查。

10. Practice Questions | 练习题目

Try these questions yourself before checking the answers below.

先自己尝试完成下面的题目，再对照后面的答案。

Simplify: 8k + 3k – 2k.
化简：8k + 3k – 2k。
Simplify: 4x + 2y – x + 3y.
化简：4x + 2y – x + 3y。
Simplify: 5m² + 2m – 3m² + 6m.
化简：5m² + 2m – 3m² + 6m。
Simplify: 7c – 3d + 2c + 4d – d.
化简：7c – 3d + 2c + 4d – d。

Answers: 9k; 3x + 5y; 2m² + 8m; 9c + 0d, which is simply 9c.

答案：9k；3x + 5y；2m² + 8m；9c + 0d，即 9c。

11. Summary | 总结

Combining like terms is a fundamental skill in algebra. Remember that like terms must have exactly the same variable part, and you only add or subtract the coefficients while keeping the variable part unchanged.

合并同类项是代数中的一项基础技能。请记住：同类项必须具有完全相同的变量部分，你只需要对系数进行加减运算，同时保持变量部分不变。

Always check your signs, never combine unlike terms, and write your working clearly. With consistent practice, this skill will become quick and automatic.

务必检查符号，绝不合并非同类项，并清晰地写出解题过程。通过持续练习，这个技能会变得快速而熟练。

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Combining Like Terms | 合并同类项