Simplifying Expressions: Collecting Like Terms | 合并同类项：化简代数式

📚 Simplifying Expressions: Collecting Like Terms | 合并同类项：化简代数式

In algebra, an expression often contains several terms that can be combined to make it shorter and easier to work with. This process is called collecting like terms.

在代数中，一个表达式往往包含多个可以合并的项。通过合并来使表达式更简洁、更便于处理，这个过程叫做合并同类项。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable parts, including the same powers. The numbers in front of the variables are called coefficients, and they can be different.

同类项是指变量部分完全相同（包括相同的指数）的项。变量前面的数字称为系数，系数可以不同。

5x and -3x are like terms because both contain the variable x to the power 1.
5x 和 -3x 是同类项，因为它们都含有变量 x 的一次幂。
4y² and y² are like terms because both contain y².
4y² 和 y² 是同类项，因为它们都含有 y²。
7ab and -2ab are like terms because both contain the product ab.
7ab 和 -2ab 是同类项，因为它们都含有乘积 ab。

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

Collecting like terms makes an expression simpler. A simpler expression is easier to evaluate, substitute into, or use in equations.

合并同类项能让表达式更简单。更简洁的表达式更容易求值、代入或用方程求解。

For example, 3x + 5x is the same as 8x. We cannot tell the total amount of x at a glance until we combine them.

例如，3x + 5x 与 8x 相等。如果不合并，我们无法一眼看出 x 的总量。

3x + 5x = 8x

This is based on the distributive property: 3x + 5x = (3 + 5)x = 8x.

这基于乘法分配律：3x + 5x = (3 + 5)x = 8x。

3. Adding and Subtracting Like Terms | 同类项的加减

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

合并同类项时，将系数相加或相减，变量部分保持不变。

Example: Simplify 7x + 2x – 4x.

示例：化简 7x + 2x – 4x。

7x + 2x – 4x = (7 + 2 – 4)x = 5x

Example: Simplify 3a – 5a + a.

示例：化简 3a – 5a + a。

3a – 5a + a = (3 – 5 + 1)a = -a

Remember that a term like a has a coefficient of 1, so a = 1a.

记住，像 a 这样的项系数为 1，即 a = 1a。

4. Combining Terms with Different Powers | 合并不同次幂的项

Terms are only like terms if the powers are the same. For example, x and x² are not like terms, so they cannot be combined.

只有当指数相同时才是同类项。例如，x 和 x² 不是同类项，所以不能合并。

Example: Simplify 2x² + 3x – x² + 5x.

示例：化简 2x² + 3x – x² + 5x。

Group the x² terms and the x terms separately:

分别组合 x² 项和 x 项：

2x² – x² + 3x + 5x = x² + 8x

Notice that x² and x are kept separate in the final answer.

注意最终答案中 x² 和 x 需要分开书写。

Example: Simplify 4y² – 2y² + 3y – 1.

示例：化简 4y² – 2y² + 3y – 1。

4y² – 2y² + 3y – 1 = 2y² + 3y – 1

5. Handling More Than One Variable | 处理多个变量

When there are multiple variables, only terms with identical variable combinations can be combined.

当出现多个变量时，只有变量组合完全相同的项才能合并。

Example: Simplify 5ab + 3a – 2ab + 7b.

示例：化简 5ab + 3a – 2ab + 7b。

Collect the ab terms, the a terms, and the b terms separately:

分别合并 ab 项、a 项和 b 项：

(5ab – 2ab) + 3a + 7b = 3ab + 3a + 7b

Notice that 3a and 7b are not like terms, so they remain as separate terms.

注意 3a 和 7b 不是同类项，所以它们保持为两个单独的项。

Example: Simplify 4x²y + 2xy – x²y + 3xy.

示例：化简 4x²y + 2xy – x²y + 3xy。

3x²y + 5xy

Here x²y and xy are different because the power of x is different.

这里 x²y 与 xy 不同，因为 x 的指数不同。

6. Working with Brackets | 括号的处理

When an expression contains brackets, first expand the brackets, then collect like terms.

当表达式中含有括号时，先去括号，再合并同类项。

Example: Simplify 2(x + 3) + 4(x – 1).

示例：化简 2(x + 3) + 4(x – 1)。

First expand:

先去括号：

2x + 6 + 4x – 4

Then collect like terms:

然后合并同类项：

2x + 4x + 6 – 4 = 6x + 2

Example: Simplify 3(2x – 5) – 2(x + 1).

示例：化简 3(2x – 5) – 2(x + 1)。

Expand carefully with the negative sign:

注意负号仔细去括号：

6x – 15 – 2x – 2 = 4x – 17

Remember that subtracting a bracket means subtracting every term inside it.

记住，减去一个括号相当于减去括号内的每一项。

7. Dealing with Negative Signs | 负号的处理

Negative coefficients are common. Always keep the sign with its term when rearranging.

负系数很常见。在重新排列时，一定要把符号和它的项一起移动。

Example: Simplify -3x + 5x – 2x.

示例：化简 -3x + 5x – 2x。

(-3 + 5 – 2)x = 0x = 0

Example: Simplify 4 – 2x + 3 – x.

示例：化简 4 – 2x + 3 – x。

Collect constant terms and x terms:

合并常数项和 x 项：

4 + 3 – 2x – x = 7 – 3x

It is conventional to write the variable term first, so we can write -3x + 7 or 7 – 3x.

通常把变量项写在前面，所以可以写成 -3x + 7 或 7 – 3x。

8. Word Problems and Perimeter | 应用题与周长

Collecting like terms is very useful in geometry. For example, the perimeter of a triangle with side lengths given in algebraic form can be simplified.

合并同类项在几何中非常有用。例如，三角形的三边长度用代数式表示时，可以把周长化简。

Example: A triangle has sides of lengths 3x + 2, 2x – 1, and x + 5. Find the perimeter.

示例：三角形三边长度分别为 3x + 2、2x – 1 和 x + 5。求周长。

Perimeter = sum of all sides:

周长 = 所有边之和：

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

So the perimeter simplifies to 6x + 6.

因此周长化简为 6x + 6。

Example: A rectangle has length 4a + b and width 2a – b. Write the perimeter in simplest form.

示例：矩形的长为 4a + b，宽为 2a – b。写出最简形式的周长。

Perimeter = 2(length + width):

周长 = 2(长 + 宽)：

2[(4a + b) + (2a – b)] = 2(6a) = 12a

Here the b terms cancel out.

这里 b 项相互抵消。

9. Common Mistakes to Avoid | 常见错误避坑

Students often make small errors when collecting like terms. Here are the most common ones and how to avoid them.

学生在合并同类项时常犯一些小错误。以下是最常见的错误及避免方法。

Mistake \| 错误	Correction \| 正确
3x + 4y = 7xy	3x + 4y cannot be simplified (not like terms)
x² + x = x³	x² + x cannot be simplified (different powers)
2a – (a – 3) = 2a – a – 3	2a – (a – 3) = 2a – a + 3 = a + 3

Always check that the variable part is identical before combining.

合并前务必检查变量部分是否完全相同。

10. Practice Questions | 练习

Try these questions yourself before checking the answers.

先自己尝试这些题目，再对照答案。

1. Simplify 6x + 3x – 2x.
2. Simplify 5a + 2b – 3a + b.
3. Simplify 4x² + 2x – x² – 5x.
4. Simplify 3(2y + 1) + 2(y – 4).
5. Simplify 7pq – 3p + 2q + pq – 5q.

Answers: 1) 7x 2) 2a + 3b 3) 3x² – 3x 4) 8y – 5 5) 8pq – 3p – 3q

答案：1) 7x 2) 2a + 3b 3) 3x² – 3x 4) 8y – 5 5) 8pq – 3p – 3q

Collecting like terms is a foundational skill for all algebra. Mastering it will make later topics such as solving equations, expanding brackets, and factorising much easier.

合并同类项是所有代数的基础技能。掌握它会使后续的方程求解、去括号、因式分解等专题变得轻松许多。

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Simplifying Expressions: Collecting Like Terms | 合并同类项：化简代数式