📚 Collecting Like Terms | 合并同类项
Algebra is the language of mathematics, and expressions are its sentences. To read and write this language fluently, you must learn to simplify expressions by collecting like terms — the algebraic equivalent of tidying up a room by grouping similar objects together.
代数是数学的语言，而表达式是它的句子。要流利地读写这门语言，你必须学会通过合并同类项来化简表达式——这就像把房间里相似的物品归类整理一样。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable part — the same variables raised to the same powers. Only the numerical coefficient may differ.
同类项是指变量部分完全相同的项——即相同的变量、相同的幂次。只有数字系数可以不同。
For example, 3x and 5x are like terms because both contain the variable x to the power 1.
例如，3x 和 5x 是同类项，因为它们都包含一次方的变量 x。
By contrast, 2x and 2x² are not like terms, because x and x² are different powers of x.
相反，2x 和 2x² 不是同类项，因为 x 和 x² 是 x 的不同幂次。
Key idea: the coefficient can be any number, but the variable part must match perfectly.
关键概念：系数可以是任意数，但变量部分必须完全相同。
2. How to Identify Like Terms | 如何识别同类项
To check whether two terms are like terms, cover up the coefficients and compare what remains.
要判断两个项是否为同类项，可以遮住系数，比较剩下的部分。
3x and 7x → both become x → like terms
3x 和 7x → 都剩下 x → 同类项
4xy and 9xy → both become xy → like terms
4xy 和 9xy → 都剩下 xy → 同类项
5x and 5x² → become x and x² → not like terms
5x 和 5x² → 剩下 x 和 x² → 不是同类项
Remember that multiplication is commutative, so xy and yx are the same — they are like terms.
记住乘法满足交换律，因此 xy 和 yx 相同——它们是同类项。
Constant terms (numbers without variables) are all like terms: 3, −7, 0.5, and 12 can all be combined.
常数项（不含变量的数字）都是同类项：3、−7、0.5 和 12 都可以合并。
3. Combining Coefficients | 合并系数
When collecting like terms, you add or subtract only the coefficients, and keep the variable part exactly unchanged.
合并同类项时，只对系数进行加减，变量部分保持不变。
3x + 5x = (3 + 5)x = 8x
For subtraction, apply the same rule: 7a − 2a = 5a.
减法同理：7a − 2a = 5a。
When the coefficient is 1, remember it is hidden: x means 1x, so x + 4x = 5x.
系数为 1 时，它是省略的：x 就是 1x，所以 x + 4x = 5x。
When the result is 0, the term disappears entirely: 3y − 3y = 0.
当结果为零时，该项完全消失：3y − 3y = 0。
4. Simplifying Expressions with Brackets | 含括号表达式的化简
Expressions often contain brackets. You must first expand the brackets (multiply out), then collect like terms.
表达式常常包含括号。你必须先展开括号（乘开），再合并同类项。
3(x + 2) + 2(2x − 1). Expand: 3x + 6 + 4x − 2. Now collect: (3x + 4x) + (6 − 2) = 7x + 4.
例如，3(x + 2) + 2(2x − 1)。展开得：3x + 6 + 4x − 2。再合并：(3x + 4x) + (6 − 2) = 7x + 4。
ting a bracket, change every sign inside the bracket: 4m − (3m − 2) = 4m − 3m + 2 = m + 2.
减去一个括号时，要改变括号内每一项的符号：4m − (3m − 2) = 4m − 3m + 2 = m + 2。
5. Expressions with Multiple Variables | 多变量表达式
In an expression with several variables, collect terms of each variable type separately.
在包含多个变量的表达式中，要分别合并每种变量类型的项。
4x + 3y − x + 2y = (4x − x) + (3y + 2y) = 3x + 5y
Notice that x and y terms are collected independently — they never mix.
注意 x 项和 y 项是各自独立合并的——它们永远不会相混。
x² + 3x + 5x² − x = 7x² + 2x. The x² terms and the x terms are treated as different types.
幂次同样如此：2x² + 3x + 5x² − x = 7x² + 2x。x² 项和 x 项被视为不同的类型。
6. Common Mistakes to Avoid | 常见错误警示
Mistake 1: Combining 2x + 2 into 4x. This is wrong because the second term has no x; they are not like terms.
错误一：把 2x + 2 合并成 4x。这是错的，因为第二项不含 x，它们不是同类项。
Mistake 2: Confusing x and x². The expression x + x² cannot be simplified any further.
错误二：混淆 x 和 x²。表达式 x + x² 不能再化简。
tracting. Always write out intermediate steps: (6y − 2) − (3y − 2) = 6y − 2 − 3y + 2 = 3y.
错误三：减法时丢失负号。务必写出中间步骤：(6y − 2) − (3y − 2) = 6y − 2 − 3y + 2 = 3y。
Mistake 4: Forgetting that −x means −1x, so −x + 4x = 3x.
错误四：忘记 −x 就是 −1x，所以 −x + 4x = 3x。
7. Worked Examples | 典型例题精解
Example A: Simplify 5a + 3b − 2a + 4b.
例题 A：化简 5a + 3b − 2a + 4b。
Group like terms: (5a − 2a) + (3b + 4b) = 3a + 7b.
分组同类项：(5a − 2a) + (3b + 4b) = 3a + 7b。
Example B: Simplify 2(3x + 1) + 4(x − 5).
例题 B：化简 2(3x + 1) + 4(x − 5)。
Expand: 6x + 2 + 4x − 20. Collect: (6x + 4x) + (2 − 20) = 10x − 18.
展开：6x + 2 + 4x − 20。合并：(6x + 4x) + (2 − 20) = 10x − 18。
Example C: Simplify 5p − (2p + 3) + 7.
例题 C：化简 5p − (2p + 3) + 7。
Remove the bracket with care: 5p − 2p − 3 + 7 = 3p + 4.
小心去括号：5p − 2p − 3 + 7 = 3p + 4。
8. Practice Questions | 巩固练习
Try these on your own before checking the answers below.
先独立完成以下练习，再核对答案。
| Question | 题目 | Answer | 答案 |
| 1. 7x + 3x | 10x |
| 2. 9y − 4y + y | 6y |
| 3. 4a + 5b − a + 2b | 3a + 7b |
| 4. 3x² + 2x + x² − 5x | 4x² − 3x |
| 5. 2(3x − 1) + 3(2 − x) | 6x − 2 + 6 − 3x = 3x + 4 |
| 6. 8m − (5m − 3) | 3m + 3 |
9. Exam Tips for IGCSE | 考试高分技巧
Tip 1: Underline or circle like terms with the same shape before combining — this prevents careless errors.
技巧一：合并前用相同标记圈出同类项——这可以防止粗心错误。
Tip 2: Always write out at least one intermediate step in your working; examiners award method marks.
技巧二：解题过程中至少写出一行中间步骤；阅卷官会给出步骤分。
Tip 3: Pay special attention to signs. The sign in front of a term belongs to that term, so treat it as part of the coefficient.
技巧三：特别注意符号。项前面的符号属于该项，应将其视为系数的一部分。
Tip 4: After simplifying, check that the number of terms in your answer makes sense — simplification should reduce the expression, not create new terms.
技巧四：化简后，检查答案的项数是否合理——化简应该减少项数，而不是产生新项。
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