📚 Collecting Like Terms | 合并同类项
In algebra, simplifying expressions is a core skill. Collecting like terms is the process of adding or subtracting terms that have the same variable part, so that an expression becomes shorter and easier to work with.
在代数中，化简表达式是一项核心技能。合并同类项就是把具有相同变量部分的项进行加减运算，让表达式变得更简短、更易于处理。
1. What are Terms? | 什么是项？
An algebraic term is a number, a variable, or a product of numbers and variables. For example, in the expression 3x + 5y − 2, the terms are 3x, 5y and −2. The sign in front of the term belongs to it.
代数项是一个数、一个变量，或数与变量的乘积。例如，在表达式 3x + 5y − 2 中，各项分别是 3x、5y 和 −2。项前面的正负号属于该项本身。
Each term has two parts: a numerical coefficient and a variable part. In the term −7ab, the coefficient is −7 and the variable part is ab.
每一项都有两个部分：数字系数和变量部分。在项 −7ab 中，系数是 −7，变量部分是 ab。
2. What are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable part, including the same powers. The coefficients can be different. For example, 4x and −2x are like terms because both have the variable x raised to the power 1.
同类项是指变量部分完全相同（包括指数也相同）的项。系数可以不同。例如，4x 和 −2x 是同类项，因为它们都含有变量 x 的一次幂。
3x² and 5x² are like terms → same variable x²
3x² and 5x are NOT like terms → different powers
3x² 与 5x² 是同类项 → 变量 x² 相同
3x² 与 5x 不是同类项 → 幂次不同
- Like terms: 7xy and −3xy
- Not like terms: 7xy and 7x (different variable parts)
- Not like terms: 2a² and 2a (different exponents)
- 同类项：7xy 与 −3xy
- 非同类项：7xy 与 7x（变量部分不同）
- 非同类项：2a² 与 2a（指数不同）
3. The Rule for Collecting Like Terms | 合并同类项的规则
To collect like terms, keep the variable part unchanged and add or subtract the coefficients. The sign in front of each coefficient must be included.
合并同类项时，保持变量部分不变，对系数进行加减运算。运算时要注意包含各项前面的正负号。
ax + bx = (a+b)x
ax − bx = (a−b)x
ax + bx = (a+b)x
ax − bx = (a−b)x
For example, 6x + 3x = (6+3)x = 9x. The variable x is simply carried through.
例如，6x + 3x = (6+3)x = 9x。变量 x 直接保留。
4. Simplifying Expressions with One Variable | 含一个变量的表达式化简
When an expression contains only one type of variable, collect all coefficients together. Always write the expression in a neat final form.
当一个表达式只含有一种变量时，将所有系数合在一起。最终结果要书写整洁。
Simplify: 5x + 8 − 3x + 2
化简：5x + 8 − 3x + 2
Step 1: Identify like terms. 5x and −3x are like; 8 and 2 are constants.
第一步：找出同类项。5x 与 −3x 是同类项；8 与 2 是常数项。
Step 2: Combine the x terms: 5x − 3x = 2x. Combine constants: 8 + 2 = 10.
第二步：合并 x 项：5x − 3x = 2x。合并常数项：8 + 2 = 10。
Answer: 2x + 10
答案：2x + 10
5. Working with Negative Coefficients | 处理负系数
Negative coefficients often appear in expressions, especially when subtracting. Be careful to apply the sign of each term correctly.
在减法中经常出现负系数。要小心地正确处理每一项的符号。
Simplify: 4a − 9a + a
化简：4a − 9a + a
Combine exactly from left to right: 4a − 9a = −5a; then −5a + a = −4a.
从左到右依次合并：4a − 9a = −5a；然后 −5a + a = −4a。
Answer: −4a
答案：−4a
6. Combining Terms with Different Variables | 合并含不同变量的项
If an expression has several different variables, such as x and y, you must collect all x terms together and all y terms together. Do not combine different variables.
如果表达式中含有多个不同的变量，例如 x 和 y，你必须要分别合并所有 x 项和所有 y 项。不可将不同变量合并。
Simplify: 7m + 4n − 2m + 3n
化简：7m + 4n − 2m + 3n
Group m terms: 7m − 2m = 5m. Group n terms: 4n + 3n = 7n.
合并 m 项：7m − 2m = 5m。合并 n 项：4n + 3n = 7n。
Answer: 5m + 7n
答案：5m + 7n
Notice that m and n cannot be added together because they are different variables.
注意：m 和 n 是不同的变量，不能相加。
7. Collecting Like Terms with Powers | 含幂次的同类项合并
When variables have exponents, you can only collect terms that share the same exponent. So x² terms combine with x² terms, but not with x terms.
当变量带有指数时，只能合并指数相同的项。因此，x² 项只能与 x² 项合并，不能与 x 项合并。
Simplify: 3x² + 2x − x² + 5x
化简：3x² + 2x − x² + 5x
Collect x² terms: 3x² − x² = 2x². Collect x terms: 2x + 5x = 7x.
合并 x² 项：3x² − x² = 2x²。合并 x 项：2x + 5x = 7x。
Answer: 2x² + 7x
答案：2x² + 7x
8. Brackets and Collecting Like Terms | 括号与合并同类项
Sometimes you may need to expand brackets before you can collect like terms. Use the distributive law: multiply the term outside the bracket by every term inside.
有时需要先去括号，然后才能合并同类项。使用乘法分配律：将括号外的项乘以括号内的每一项。
Simplify: 3(2x + 4) + 5(x − 2)
化简：3(2x + 4) + 5(x − 2)
First expand: 3(2x + 4) = 6x + 12; 5(x − 2) = 5x − 10.
先去括号：3(2x + 4) = 6x + 12；5(x − 2) = 5x − 10。
Then collect: 6x + 5x = 11x; 12 − 10 = 2.
然后合并：6x + 5x = 11x；12 − 10 = 2。
Answer: 11x + 2
答案：11x + 2
9. Common Mistakes to Avoid | 常见错误与避免方法
Many students make predictable errors when collecting like terms. Knowing these will help you avoid them in exams.
许多学生在合并同类项时会犯一些常见错误。了解这些错误有助于你在考试中避免它们。
- Mistake: combining x and x². Remember, x and x² are not like terms.
- Mistake: ignoring negative signs. For example, −3y + 5y = 2y, not 8y.
- Mistake: changing the variable. When you combine 4ab + 3ab, the result is 7ab, not 7a²b².
- Mistake: mixing constants and variables. 5 + 2x stays as 5 + 2x, not 7x.
- 错误：将 x 与 x² 合并。注意：x 和 x² 不是同类项。
- 错误：忽略负号。例如，−3y + 5y = 2y，而不是 8y。
- 错误：改变变量。4ab + 3ab = 7ab，而不是 7a²b²。
- 错误：混淆常数项与变量项。5 + 2x 仍是 5 + 2x，而不是 7x。
10. Practice Problems | 练习题目
Try these IGCSE-style questions by yourself before checking the solutions.
请先独立尝试这些 IGCSE 风格的题目，再对照答案。
- Simplify 6x + 3x − 2x
- Simplify 5a + 2b − a + 4b
- Simplify 4p − 7p + 2p
- Simplify 2m² + 3m − m² + 5m
- Simplify 4(2y + 1) − 3(y − 2)
- 化简 6x + 3x − 2x
- 化简 5a + 2b − a + 4b
- 化简 4p − 7p + 2p
- 化简 2m² + 3m − m² + 5m
- 化简 4(2y + 1) − 3(y − 2)
11. Solutions | 答案解析
Check your answers step by step.
逐步对照答案。
1. 6x + 3x − 2x = (6+3−2)x = 7x
2. 5a + 2b − a + 4b = (5a − a) + (2b + 4b) = 4a + 6b
3. 4p − 7p + 2p = (4−7+2)p = −1p = −p
4. 2m² + 3m − m² + 5m = (2m² − m²) + (3m + 5m) = m² + 8m
5. 4(2y + 1) − 3(y − 2) = 8y + 4 − 3y + 6 = (8y − 3y) + (4 + 6) = 5y + 10
1. 6x + 3x − 2x = (6+3−2)x = 7x
2. 5a + 2b − a + 4b = (5a − a) + (2b + 4b) = 4a + 6b
3. 4p − 7p + 2p = (4−7+2)p = −1p = −p
4. 2m² + 3m − m² + 5m = (2m² − m²) + (3m + 5m) = m² + 8m
5. 4(2y + 1) − 3(y − 2) = 8y + 4 − 3y + 6 = (8y − 3y) + (4 + 6) = 5y + 10
12. Exam Tips | 考试提示
In IGCSE exams, collecting like terms often appears as part of a longer algebra question. Always write your final answer in alphabetical order when possible.
在 IGCSE 考试中，合并同类项通常作为较长代数题的一部分。最终答案中，变量尽量按字母顺序排列。
- Always look for like terms before doing any other operation.
- Use a rough workspace to group terms visually.
- Check that you have every term exactly once.
- If a term is alone, keep it in the answer.
- 进行其他运算之前，先找出同类项。
- 在草稿纸上用划线或分组的方式将同类项标记出来。
- 检查每一项是否都被完整用上。
- 如果某一项没有同类项，直接保留在答案中。
Mastering collecting like terms will make algebra expansion, solving equations, and simplifying fractions much easier. Practice daily until it becomes automatic.
掌握合并同类项，会让后续的展开、解方程和化简分式变得更加容易。每天练习，直至熟练自如。
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