📚 Combining Like Terms | 合并同类项
In IGCSE Mathematics, algebra often asks you to "simplify" an expression. The first and most important tool is collecting or combining like terms. This skill turns long, messy expressions into short, clear ones and is essential for solving equations, factorising, and working with formulas.
在 IGCSE 数学中，代数题常常要求你“化简”一个表达式。第一个也是最重要的工具就是合并同类项。这项技能能把又长又乱的表达式变成简短清晰的形式，是解方程、因式分解和运用公式的基础。
1. What Are Terms in Algebra? | 代数中的项是什么？
A term is a single number, a single variable, or numbers and variables multiplied together. In an expression, terms are separated by plus or minus signs. For example, 5x² − 3x + 7 has three terms: 5x², −3x, and 7.
项是一个单独的数、一个单独的变量，或者数与变量相乘的结果。在表达式中，各项由加号或减号分开。例如 5x² − 3x + 7 有三项：5x²、−3x 和 7。
| Expression | Terms |
| 4x + 9 | 4x and 9 |
| 2a − 3b + 5c | 2a, −3b, and 5c |
| x² + 2xy − y² | x², 2xy, and −y² |
2. Identifying Like Terms | 识别同类项
Like terms have exactly the same variable part, including the same powers. The coefficients can be different. For example, 2x and −5x are like terms because both have x. Similarly, 3x² and 7x² are like terms because both have x². However, 4x and 4x² are not like terms because the powers of x are different.
同类项必须有完全相同的字母部分，包括相同的指数。系数可以不同。例如 2x 和 −5x 是同类项，因为都含有 x。同样，3x² 和 7x² 是同类项，因为都含有 x²。但 4x 和 4x² 不是同类项，因为 x 的指数不同。
| Pair | Like terms? |
| 3x and 8x | Yes |
| 2x² and −6x² | Yes |
| 4y and 4y² | No |
| 5ab and 5a | No |
3. Coefficients and Constants | 系数与常数项
The number in front of a variable is called the coefficient. In 7y, 7 is the coefficient. In −2a, −2 is the coefficient. If no number is written, the coefficient is 1, so x means 1x. A constant term is a number on its own with no variable, such as 9 in 3x + 9.
变量前面的数字叫做系数。在 7y 中，7 是系数。在 −2a 中，−2 是系数。如果没有写出数字，系数就是 1，因此 x 表示 1x。常数项是没有变量的独立数字，比如 3x + 9 中的 9。
term = coefficient × variable
−2a = −2 × a
4. The Golden Rule of Combining | 合并的黄金法则
cause 3 + 2 = 5 while the x stays the same. Unlike terms cannot be combined into one term.
你只能对同类项进行加减。合并时，保持字母部分不变，只对系数进行加减。例如 3x + 2x = 5x，因为 3 + 2 = 5，而 x 保持不变。不同类项不能合并为一项。
ax + bx = (a + b)x
ax − bx = (a − b)x
- 3x + 2x = 5x
- 7y − 4y = 3y
- 2x + 3y cannot be combined
5. Step-by-Step Simplification | 逐步化简
Step 1: Identify terms and their signs. Step 2: Group like terms together. Step 3: Combine coefficients. Step 4: Write the simplified expression, usually with terms in descending powers of the variable.
步骤一：识别各项及其符号。步骤二：把同类项分组。步骤三：合并系数。步骤四：写出化简后的表达式，通常按变量幂次从高到低排列。
Simplify 4x + 3y − 2x + y.
化简 4x + 3y − 2x + y。
4x + 3y − 2x + y = (4x − 2x) + (3y + y) = 2x + 4y
- Group the x terms: 4x − 2x = 2x
- Group the y terms: 3y + y = 4y
- Final answer: 2x + 4y
6. Combining Terms with Multiple Variables | 合并含多个变量的项
t. Also, 3xy and 3yx are like terms because multiplication order does not matter: xy = yx.
当项含有多个变量时，字母部分必须完全一致。4ab 和 2ab 是同类项，但 4ab 和 4a 不是。另外，3xy 和 3yx 是同类项，因为乘法顺序不影响结果：xy = yx。
5xy + 2yx − 3xy = (5 + 2 − 3)xy = 4xy
Remember that x²y and xy² are not like terms because the powers are attached to different variables.
记住，x²y 和 xy² 不是同类项，因为幂次属于不同的变量。
7. Removing Brackets Before Combining | 先去括号再合并
utive law. Then combine any like terms. For example, 2(x + 3) + 3x = 2x + 6 + 3x = 5x + 6.
如果表达式含有括号，先用分配律展开括号，然后合并同类项。例如，2(x + 3) + 3x = 2x + 6 + 3x = 5x + 6。
ust be distributed to every term inside the bracket: 5x − 2(x − 1) = 5x − 2x + 2 = 3x + 2.
遇到减法时，符号必须分配给括号内的每一项：5x − 2(x − 1) = 5x − 2x + 2 = 3x + 2。
a(b + c) = ab + ac
8. Common Mistakes to Avoid | 常见错误与避免方法
Common mistakes include adding unlike terms, changing the variable part, forgetting the sign in front of a term, and incorrectly combining constants with variables. Always group signs with terms when rearranging.
常见错误包括把不同类项相加、改变了字母部分、忘记项前面的符号，以及把常数项和变量错误合并。重新排列时，一定要把符号和项放在一起处理。
- Wrong: x + x² = 2x² | Correct: x + x² cannot be simplified
- Wrong: 3x + 2 = 5x | Correct: 3x + 2 cannot be simplified
- Wrong: 4x − x = 4 | Correct: 4x − x = 3x
9. Worked Examples | 典型例题
Example 1: Simplify 6a + 2b − 3a + 5b.
例 1：化简 6a + 2b − 3a + 5b。
(6a − 3a) + (2b + 5b) = 3a + 7b
Example 2: Simplify 4x² + 3x − 2x² + x.
例 2：化简 4x² + 3x − 2x² + x。
(4x² − 2x²) + (3x + x) = 2x² + 4x
Example 3: Simplify 3(m + 2) − 2m + 4.
例 3：化简 3(m + 2) − 2m + 4。
3m + 6 − 2m + 4 = (3m − 2m) + (6 + 4) = m + 10
10. Practice and Exam Tips | 练习与考试技巧
In IGCSE exams, show your grouping steps to gain method marks, even if you make an arithmetic slip. Write like terms clearly, use a highlighter if allowed, and check your answer by substituting a simple value for the variable. Always present the final answer in standard form with terms in descending powers.
在 IGCSE 考试中，写出分组步骤可以拿到方法分，即使计算有小失误。清楚标出同类项，如果允许可以使用荧光笔，并通过代入一个简单数值检查答案。最后答案要按标准形式书写，即按幂次从高到低排列。
- Simplify: 7p + 3q − 4p + q (Answer: 3p + 4q)
- Simplify: 5x + 2(x − 3) (Answer: 7x − 6)
- Simplify: 2xy + 3x − xy + 4x (Answer: xy + 7x)
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