Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain several terms that share the same variable part. Collecting like terms means simplifying an expression by adding or subtracting these similar terms, which makes the expression shorter and easier to work with.

在代数中，表达式常常包含多个具有相同变量部分的项。合并同类项就是通过加减这些相似的项来化简表达式，使其更简短、更易于处理。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable factors, raised to the same powers. The coefficients can be different, but the variable part must match exactly.

同类项是指具有完全相同变量因子且幂次也相同的项。系数可以不同，但变量部分必须完全匹配。

Example: 3x and 5x are like terms because both contain the variable x.

例如：3x 和 5x 是同类项，因为两者都含有变量 x。
Example: 4x² and 7x² are like terms, but 4x² and 7x are not like terms because the powers of x are different.

例如：4x² 和 7x² 是同类项，但 4x² 和 7x 不是同类项，因为 x 的幂次不同。

Constant terms (numbers without variables) are also like terms with each other.

常数项（不含变量的数字）彼此之间也是同类项。

2. Identifying Like Terms | 识别同类项

To identify like terms, look at the variable part of each term. The coefficient (the number in front of the variable) does not affect whether terms are like.

要识别同类项，请观察每一项的变量部分。系数（变量前的数字）不影响项是否为同类项。

Expression	Like terms	Not like terms
6a + 2a – 3b	6a and 2a	3b (different variable)
5xy + 8x – 2xy	5xy and -2xy	8x (missing y)
3p² + 4p + p²	3p² and p²	4p (different power)

Remember that a term like ab is the same as ba, so ab and ba are like terms.

请记住，ab 与 ba 表示相同的乘积，因此 ab 和 ba 是同类项。

3. The Distributive Law and Combining | 分配律与合并

Collecting like terms relies on the distributive law: ax + bx = (a + b)x. This works because x is a common factor.

合并同类项依据分配律：ax + bx = (a + b)x。这是因为 x 是公因子。

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

Similarly, 2y + 3y – y = (2 + 3 – 1)y = 4y.

类似地，2y + 3y – y = (2 + 3 – 1)y = 4y。

This process is also called “combining like terms” or “simplifying algebraic expressions.”

这个过程也称为“合并同类项”或“化简代数表达式”。

4. Combining with Different Coefficients | 合并不同系数

When coefficients are integers, simply add or subtract them according to the signs in front of each term.

当系数为整数时，只需根据各项前面的符号对它们进行加减。

7x – 2x + 3x = (7 – 2 + 3)x = 8x

Always include the sign of the term when adding coefficients. For example, in 4x – 6x, subtract 6 from 4 to get -2x.

合并系数时始终带上项的符号。例如，4x – 6x，用4减去6得到 -2x。

Example: 5m + 2m – 8m = (5 + 2 – 8)m = -1m = -m

例：5m + 2m – 8m = (5 + 2 – 8)m = -1m = -m
Example: -3a + 9a + a = (-3 + 9 + 1)a = 7a

例：-3a + 9a + a = (-3 + 9 + 1)a = 7a

5. Dealing with Negative Numbers | 处理负数

Negative coefficients can be tricky. Remember the rules for adding and subtracting integers when you collect like terms.

负系数可能容易出错。合并同类项时，请记住整数加减法的规则。

2x – (-3x) = 2x + 3x = 5x

If an expression has subtraction before a term, treat that term as negative. For example, 8y – 5y means 8y + (-5y) = 3y.

如果项前面是减法，则将该项视为负数。例如，8y – 5y 表示 8y + (-5y) = 3y。

Expression	Result
4x – 7x	-3x
-2y – 5y	-7y
-6a + 2a	-4a

6. Combining Multiple Terms | 合并多项

Often an expression contains several groups of like terms. Group the terms by their variable part, then combine each group separately.

表达式通常包含几组同类项。按变量部分分组，然后分别合并每组。

3a + 4b – 2a + 7b = (3a – 2a) + (4b + 7b) = a + 11b

Pick out each different variable and add its coefficients together. Write the simplified expression without changing the order of the variables.

找出每个不同的变量，将它们的系数相加。写出化简后的表达式，不要改变变量的顺序。

Example: 5x² + 3x – 2x² + 6x = (5x² – 2x²) + (3x + 6x) = 3x² + 9x

例：5x² + 3x – 2x² + 6x = (5x² – 2x²) + (3x + 6x) = 3x² + 9x
Example: 2xy + 3x – xy + 4y = (2xy – xy) + 3x + 4y = xy + 3x + 4y

例：2xy + 3x – xy + 4y = (2xy – xy) + 3x + 4y = xy + 3x + 4y

7. Fractions and Decimals | 分数与小数

Like terms with fractional or decimal coefficients are combined using the same rules. Use a common denominator when adding fractions.

含有分数或小数系数的同类项使用相同的规则合并。分数相加时需要公分母。

(3/4)x + (1/4)x = (3/4 + 1/4)x = (4/4)x = x

Decimal coefficients are added just like ordinary decimals.

小数系数按普通小数相加。

Example: 0.5y + 1.2y = (0.5 + 1.2)y = 1.7y

例：0.5y + 1.2y = (0.5 + 1.2)y = 1.7y
Example: (1/2)m – (1/3)m = (3/6 – 2/6)m = (1/6)m

例：(1/2)m – (1/3)m = (3/6 – 2/6)m = (1/6)m

8. Expanding Brackets before Combining | 先去括号再合并

If an expression contains brackets, you must expand the brackets before you can collect like terms.

如果表达式中包含括号，必须先展开括号，然后才能合并同类项。

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

Use the distributive law to multiply each term inside the bracket by the term outside.

使用分配律将括号外的项乘以括号内的每一项。

Example: 3(2a – 1) + 5a = 6a – 3 + 5a = 11a – 3

例：3(2a – 1) + 5a = 6a – 3 + 5a = 11a – 3
Example: 2(3x + 4) – 3(2x – 1) = 6x + 8 – 6x + 3 = 11

例：2(3x + 4) – 3(2x – 1) = 6x + 8 – 6x + 3 = 11

9. Word Problems and Applications | 应用题

Collecting like terms is used to simplify perimeter expressions, total cost problems, and geometric formulas.

合并同类项常用于化简周长表达式、总成本问题以及几何公式。

For a rectangle with length x + 3 and width x – 1, the perimeter is 2(x + 3) + 2(x – 1). Expanding and collecting gives:

例如，一个长方形的长为 x + 3，宽为 x – 1，其周长为 2(x + 3) + 2(x – 1)。展开并合并得到：

2x + 6 + 2x – 2 = 4x + 4

So the perimeter simplifies to 4x + 4. This is much easier to use in further calculations.

因此周长化简为 4x + 4。这在进一步计算中更加方便。

10. Common Mistakes to Avoid | 常见错误

Students often make mistakes when variables have exponents or when terms look similar but are not like terms.

当变量带有指数或项看起来相似但并非同类项时，学生常犯错误。

Mistake	Why it is wrong	Correct method
3x + 2x² = 5x³	Different powers; cannot be combined	3x + 2x² stays as 3x + 2x²
4(x + 2) = 4x + 2	Did not multiply 2 by 4	4(x + 2) = 4x + 8
5a – (a – 3) = 5a – a – 3	Subtraction sign must apply to all terms inside	5a – (a – 3) = 5a – a + 3 = 4a + 3

Always check whether each term has the same variable part, including exponents, before combining.

合并之前一定要检查每一项是否具有相同的变量部分，包括指数。

11. Practice Problems | 练习

Try these problems to master collecting like terms.

尝试以下练习以掌握合并同类项。

1. Simplify: 7x + 2x – 5x

1. 化简：7x + 2x – 5x
2. Simplify: 3a² + 4a – a² + 6a

2. 化简：3a² + 4a – a² + 6a
3. Simplify: 2(3y – 1) + 4y

3. 化简：2(3y – 1) + 4y
4. Simplify: 5m + 2n – 3m + 7n

4. 化简：5m + 2n – 3m + 7n

Answers: 1. 4x; 2. 2a² + 10a; 3. 10y – 2; 4. 2m + 9n

答案：1. 4x；2. 2a² + 10a；3. 10y – 2；4. 2m + 9n

12. Summary | 总结

Collecting like terms is a fundamental skill in algebra. Identify terms with the same variable part, combine their coefficients, and keep different variable parts separate.

合并同类项是代数中的基础技能。识别具有相同变量部分的项，合并它们的系数，并保持不同变量部分分开。

Remember to expand brackets first, handle negative signs carefully, and never combine terms with different powers or variables.

记住先展开括号，小心处理负号，并且绝不合并不同幂次或不同变量的项。

With practice, this process becomes quick and automatic, helping you solve much larger problems confidently.

通过练习，这个过程会变得快速而自然，帮助你自信地解决更大的问题。

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Collecting Like Terms | 合并同类项