📚 Simplifying Expressions by Collecting Like Terms | 合并同类项：化简代数式
Collecting like terms is one of the most fundamental skills in algebra. It allows us to simplify messy expressions into cleaner, shorter forms — an essential first step before solving equations, substituting values, or rearranging formulas.
合并同类项是代数中最基础的技能之一。它让我们能把冗长的表达式化为更简洁的形式——这是在解方程、代值计算或重排公式之前必不可少的第一步。
1. What Are "Terms" and "Like Terms"? | 什么是"项"和"同类项"？
A term is a single 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.
项是一个单独的数字、一个变量，或者数字与变量的乘积。例如，在表达式 3x + 5y − 2 中，各项分别是 3x、5y 和 −2。
Like terms are terms that have exactly the same variable part (the same letters raised to the same powers). Only the numerical coefficient may differ.
同类项是指变量部分完全相同（相同的字母、相同的指数）的项。只有数字系数可以不同。
3x and 7x → like terms ✓
3x and 3y → not like terms ✗
4x² and 4x → not like terms ✗
Notice that x² and x are NOT like terms, because the variable part is x² for one and x for the other. The power matters.
注意：x² 和 x 不是同类项，因为它们的变量部分分别是 x² 和 x。指数很重要。
2. Why Do We Collect Like Terms? | 为什么要合并同类项？
Imagine you have 3 apples and then someone gives you 5 more apples: you now have 8 apples. But if you were given 5 oranges instead, you cannot combine them into "8 apples" — you must keep apples and oranges separate.
想象你有 3 个苹果，有人又给你 5 个苹果：你一共有 8 个苹果。但如果别人给你的是 5 个橙子，你就不能把它们算成"8 个苹果"——苹果和橙子必须分开计算。
Algebra works exactly the same way. Terms with different variable parts cannot be added together into a single term. Collecting like terms is the process of grouping and adding the terms that share the same variable part.
代数的道理完全一样。变量部分不同的项不能合并成一项。合并同类项就是将变量部分相同的项分组、相加的过程。
3. The Steps to Collect Like Terms | 合并同类项的步骤
Follow these three steps every time you simplify an expression.
每次化简表达式时，都按以下三个步骤操作。
- Step 1: Identify all the like terms by looking at the variable part.
- 第一步：通过变量部分辨认所有同类项。
- Step 2: Group them together (you can underline or colour-code them).
- 第二步：把它们归到一起（可以用下划线或颜色标记）。
- Step 3: Add or subtract the coefficients, and keep the variable part unchanged.
- 第三步：合并系数（相加或相减），变量部分保持不变。
roup the x terms: 4x − 2x = 2x. Group the constants: 7 + 3 = 10. So the answer is 2x + 10.
例如，化简 4x + 7 − 2x + 3。先把 x 项放在一起：4x − 2x = 2x；再把常数放在一起：7 + 3 = 10。所以答案是 2x + 10。
4. Working with Positive and Negative Coefficients | 处理正系数与负系数
Coefficients can be positive or negative. When collecting like terms, treat each term with its sign — the sign in front of the term belongs to that term.
系数可以是正数或负数。合并同类项时，每一项连同它前面的符号一起处理——项前面的正负号属于该项。
implify 5a − 8a. The coefficients are 5 and −8. Add them: 5 + (−8) = −3. So 5a − 8a = −3a.
化简 5a − 8a：系数为 5 和 −8。5 + (−8) = −3，所以 5a − 8a = −3a。
Simplify 6b − 4b − 2b. Coefficients: 6 + (−4) + (−2) = 0. So 6b − 4b − 2b = 0.
化简 6b − 4b − 2b：系数为 6 + (−4) + (−2) = 0，所以结果为 0。
Be careful: if the coefficient is 0, the term disappears entirely from the expression.
注意：如果合并后系数为 0，该项就完全从表达式中消失。
5. Simplifying Expressions with Multiple Variable Types | 化简含多种变量的表达式
Many expressions contain more than one type of variable. You must collect each type separately.
很多表达式包含不止一种变量。你需要分别合并每一种变量。
x − y. Collect x terms: 3x + 5x = 8x. Collect y terms: 2y − 1y = 1y. The answer is 8x + y.
化简 3x + 2y + 5x − y：x 项系数 3 + 5 = 8；y 项系数 2 − 1 = 1，所以答案为 8x + y。
p + 4q − 5. p terms: 2p − p = 1p. q terms: 3q + 4q = 7q. Constant: −5. Answer: p + 7q − 5.
化简 2p + 3q − p + 4q − 5：p 项：2p − p = p；q 项：3q + 4q = 7q；常数项：−5。答案是 p + 7q − 5。
Remember to write the terms in alphabetical order if possible — it makes your answer look more professional and easier to check.
记得尽可能按字母顺序排列各项——这样答案更规范，也更方便检查。
6. The Distributive Law: Expanding Before Collecting | 分配律：先展开再合并
Sometimes the expression contains brackets. Before you can collect like terms, you must first expand the brackets using the distributive law.
有时表达式中含有括号。在合并同类项之前，必须先利用分配律把括号展开。
a(b + c) = ab + ac. Every term inside the bracket must be multiplied by the term outside.
分配律：a(b + c) = ab + ac。括号内的每一项都必须乘以括号外的项。
3(2x + 1) = 3 × 2x + 3 × 1 = 6x + 3
− 6. Then collect x terms: 6x + 8x = 14x. Collect constants: 3 − 6 = −3. Answer: 14x − 3.
化简 3(2x + 1) + 2(4x − 3)：先展开得 6x + 3 + 8x − 6；再合并 x 项：6x + 8x = 14x；合并常数：3 − 6 = −3。答案是 14x − 3。
7. Negative Signs Outside Brackets | 括号前有负号怎么办
A negative sign outside a bracket means you multiply every term inside by −1. This reverses the sign of every term inside the bracket.
括号前有负号，意味着括号内每一项都要乘以 −1，即每一项的符号都要变号。
−(2x − 5) = −1 × 2x + (−1) × (−5) = −2x + 5
Simplify 5x − (2x − 3). Note: −(2x − 3) = −2x + 3. So 5x − 2x + 3 = 3x + 3.
化简 5x − (2x − 3)：注意 −(2x − 3) = −2x + 3，所以 5x − 2x + 3 = 3x + 3。
and: 8x − 4 − 6x − 10. Collect x: 8x − 6x = 2x. Constants: −4 − 10 = −14. Answer: 2x − 14.
化简 4(2x − 1) − 2(3x + 5)：展开得 8x − 4 − 6x − 10；x 项合并：8x − 6x = 2x；常数合并：−4 − 10 = −14。答案是 2x − 14。
8. Nested Brackets | 嵌套括号
Some questions contain brackets inside brackets. The key rule is: expand from the inside out.
有些题目含有括号套括号的情况。关键规则是：由内向外逐层展开。
2[3x − (2x − 4)]
inner bracket: 3x − (2x − 4) = 3x − 2x + 4 = x + 4. Then multiply by 2: 2(x + 4) = 2x + 8.
先处理内层括号：3x − (2x − 4) = 3x − 2x + 4 = x + 4；再乘以 2，得到 2x + 8。
+ 5] = 3a + 15. Now the full expression: 2(3a − 4) − 3a − 15 = 6a − 8 − 3a − 15 = 3a − 23.
展开 2(3a − 4) − 3[2a − (a − 5)]：内层 a − 5，所以 2a − (a − 5) = 2a − a + 5 = a + 5；外乘 3 得 3a + 15。完整表达式为 2(3a − 4) − 3a − 15 = 6a − 8 − 3a − 15 = 3a − 23。
Tip: Rewrite the expression after each stage. Do not try to skip steps in your head.
小贴士：每一步展开后都要重新写一遍表达式。不要试图在脑子里跳过步骤。
9. Working with Fractional Coefficients | 分数系数
Fractions are common in IGCSE questions. Collect like terms exactly the same way — just be careful with arithmetic.
分数在 IGCSE 考试中很常见。合并方法完全一样——只要细心做分数运算即可。
½x + ⅓x = (½ + ⅓)x = 5/6 x
Simplify 3/4 y − 1/2 y. Coefficient: 3/4 − 1/2 = 3/4 − 2/4 = 1/4. So the answer is ¼y.
化简 3/4 y − 1/2 y：系数为 3/4 − 1/2 = 3/4 − 2/4 = 1/4，所以答案为 ¼y。
When subtracting a fractional algebraic term, give both fractions the same denominator before proceeding.
当分数代数项相减时，先通分（化为相同分母）再进行计算。
10. Indices in Like Terms | 含指数的同类项
Remember that x, x², and x³ are different types of terms. They cannot be combined.
记住：x、x²、x³ 是不同类型的项，不能合并。
ify 4x² + 3x − 2x² + x. x² terms: 4x² − 2x² = 2x². x terms: 3x + x = 4x. Answer: 2x² + 4x.
化简 4x² + 3x − 2x² + x：x² 项合并：4x² − 2x² = 2x²；x 项合并：3x + x = 4x。答案为 2x² + 4x。
5x³ − x³ + 2x² − x². x³ terms: 5x³ − x³ = 4x³. x² terms: 2x² − x² = x². Answer: 4x³ + x².
化简 5x³ − x³ + 2x² − x²：x³ 项合并为 4x³；x² 项合并为 x²。答案为 4x³ + x²。
Do not confuse x² with 2x. x² means x multiplied by itself; 2x means 2 multiplied by x — they are completely different.
不要把 x² 与 2x 混淆。x² 表示 x 乘以自身；2x 表示 2 乘以 x——两者完全不同。
11. Common Exam Mistakes and How to Avoid Them | 考试常见错误及避免方法
Here are the most common traps students fall into when collecting like terms.
以下是学生在合并同类项时最常见的陷阱。
| Mistake | 错误 | Example | 示例 | Correction | 修正 |
| Adding coefficients and variables incorrectly | 错误地把系数与变量相加 | 3x + 2x = 5x² ✗ | 3x + 2x = 5x ✓ |
| Combining x with x² | 把 x 与 x² 合并 | 2x + 3x² = 5x² ✗ | 2x + 3x² cannot be simplified ✓ |
| Ignoring negative signs | 忽略负号 | 5x − 3x = 8x ✗ | 5x − 3x = 2x ✓ |
| Mistaking the sign distribution across a bracket | 括号前符号分配出错 | −(2x − 3) = −2x − 3 ✗ | −(2x − 3) = −2x + 3 ✓ |
Always double-check the sign in front of each term before combining. The most common source of error is not the addition — it is the sign.
合并前一定要仔细检查每项前面的符号。最常见的错误来源不是加法本身，而是符号问题。
12. Practice Questions and Quick Check | 练习与快速自测
Try these five questions yourself before scrolling to the answers.
先自己试做以下五道题，再对照答案。
- Q1: Simplify 7x + 3x − 5x.
- 练习1：化简 7x + 3x − 5x。
- Q2: Simplify 4a + 5b − 2a + 3b.
- 练习2：化简 4a + 5b − 2a + 3b。
- Q3: Simplify 3(2x + 1) + (x − 4).
- 练习3：化简 3(2x + 1) + (x − 4)。
- Q4: Simplify 2x² + 3x − x² − 7x.
- 练习4：化简 2x² + 3x − x² − 7x。
- Q5: Simplify 5 − 2(3x − 4) + x.
- 练习5：化简 5 − 2(3x − 4) + x。
Answers: A1: 5x. A2: 2a + 8b. A3: 7x − 1. A4: x² − 4x. A5: 13 − 5x.
答案：A1：5x；A2：2a + 8b；A3：7x − 1；A4：x² − 4x；A5：13 − 5x。
If you got all five correct, congratulations — you have mastered collecting like terms for IGCSE.
如果五道题全部做对，恭喜你——你已经掌握了 IGCSE 合并同类项的核心内容。
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