📚 Collecting Like Terms | 合并同类项
Algebraic expressions in IGCSE often contain several terms that can be simplified. Collecting like terms is the foundation of simplifying expressions, solving equations, and rearranging formulae.
IGCSE 代数表达式经常包含几个可以化简的项。合并同类项是化简表达式、解方程和变形公式的基础。
1. What Are Like Terms? | 什么是同类项
A term is a single number, a variable, or a product of numbers and variables. For example, 7, x, 4y, −2ab, and 3x² are all terms.
项是一个单独的数、一个变量，或数与变量的乘积。例如 7、x、4y、−2ab 和 3x² 都是项。
Like terms have exactly the same variable parts, including the same powers. 5x and −3x are like terms, but 5x and 5x² are not like terms.
同类项的变量部分必须完全相同，包括指数也相同。5x 和 −3x 是同类项，但 5x 和 5x² 不是同类项。
| Term | Like term | Not like term |
|---|---|---|
| 6x | −2x | 6y, 6x² |
| 4xy | −xy | 4x, 4y, 2x²y |
| −3p² | 5p² | 3p, 2p³ |
2. Why Collect Like Terms? | 为什么要合并同类项
An expression such as 3a + 5a can be written as 8a because the two terms represent the same type of object. Collecting makes expressions shorter and reduces the chance of errors in later working.
像 3a + 5a 这样的表达式可以写成 8a，因为这两项表示同一种对象。合并同类项能让表达式更短，减少后续计算中的错误。
3a + 5a = 8a
This process is used constantly in solving equations, drawing graphs, and simplifying formulae.
这一过程在解方程、画图像和化简公式中都会频繁使用。
3. The Golden Rule: Same Variable, Same Power | 黄金法则：变量相同且指数相同
Only the coefficients, the number parts, are added or subtracted. The variable part stays unchanged. For example, 7x + 2x = 9x, but 7x + 2x² cannot be simplified to 9x².
只有系数（数字部分）才相加或相减。变量部分保持不变。例如 7x + 2x = 9x，但 7x + 2x² 不能化简为 9x²。
7x + 2x = 9x, but x + x² cannot be combined
All constant numbers are like terms with each other, so 5 and −12 combine to give −7.
所有常数项都互为同类项，因此 5 和 −12 合并后得到 −7。
4. Combining Positive and Negative Terms | 合并正项与负项
Always look at the sign in front of each term. In 5x − 3x + 2, the second term is −3x, so the coefficient is −3. Therefore 5x − 3x = 2x.
一定要看每一项前面的符号。在 5x − 3x + 2 中，第二项是 −3x，所以系数是 −3。因此 5x − 3x = 2x。
5x − 3x + 2 = 2x + 2
When combining, think of the sign as being attached to the term. This prevents errors such as 5x − 3x = 8x.
合并时要把符号看作属于这一项。这能避免诸如 5x − 3x = 8x 的错误。
5. Worked Example 1: Linear Expression | 例题1：一次表达式
Simplify 4x + 3y − 2x + y.
化简 4x + 3y − 2x + y。
First collect the x terms: 4x − 2x = 2x. Then collect the y terms: 3y + y = 4y. The answer is 2x + 4y.
首先合并 x 项：4x − 2x = 2x。再合并 y 项：3y + y = 4y。答案是 2x + 4y。
4x + 3y − 2x + y = 2x + 4y
6. Worked Example 2: Mixed Powers | 例题2：混合指数
Simplify 5p² + 3p − 2p² + 4p.
化简 5p² + 3p − 2p² + 4p。
The terms 5p² and −2p² are like terms because both have p². The terms 3p and 4p are like terms because both have p. They cannot all be mixed together.
5p² 和 −2p² 是同类项，因为都有 p²。3p 和 4p 是同类项，因为都有 p。它们不能混在一起合并。
So 5p² − 2p² = 3p² and 3p + 4p = 7p, giving 3p² + 7p.
所以 5p² − 2p² = 3p²，3p + 4p = 7p，得到 3p² + 7p。
5p² + 3p − 2p² + 4p = 3p² + 7p
7. Expressions with Brackets | 含括号的表达式
If brackets appear, expand them first using the distributive law. Then collect like terms.
如果出现括号，要先用分配律展开括号，然后再合并同类项。
Simplify 2(x + 3) + 3(x − 1). Expanding gives 2x + 6 + 3x − 3. Collect the x terms and the constants: 2x + 3x = 5x and 6 − 3 = 3. The answer is 5x + 3.
化简 2(x + 3) + 3(x − 1)。展开得到 2x + 6 + 3x − 3。合并 x 项和常数项：2x + 3x = 5x，6 − 3 = 3。答案是 5x + 3。
2(x + 3) + 3(x − 1) = 5x + 3
8. Fractions and Negative Coefficients | 分数系数与负系数
Collecting works the same with fractions and negatives. For example, ½x + x means ½x + 1x, so the result is 1½x or 3x/2.
合并同类项对分数和负数同样适用。例如 ½x + x 表示 ½x + 1x，结果是 1½x 或 3x/2。
½x + x = 3x/2
Similarly, −4y + 2y = −2y, because −4 + 2 = −2. Keep the variable y unchanged.
同样，−4y + 2y = −2y，因为 −4 + 2 = −2。变量 y 保持不变。
−4y + 2y = −2y
9. Common Mistakes to Avoid | 常见错误
Do not combine terms with different powers. For example, 3x² + 2x is not 5x² or 5x; it is already in its simplest form unless other like terms exist.
不要把指数不同的项合并。例如 3x² + 2x 不等于 5x² 也不等于 5x；如果没有其他同类项，它已经是最简形式。
Always carry the sign in front of a term. In 9 − 4x + 2, the term 4x is negative, so the x term remains −4x.
要始终带着项前面的符号。在 9 − 4x + 2 中，4x 这一项是负的，所以 x 项仍为 −4x。
Do not change a variable power when adding coefficients. 2x² + 3x² = 5x², not 5x⁴.
合并系数时不要改变变量的指数。2x² + 3x² = 5x²，而不是 5x⁴。
10. Exam-Style Practice | 考试风格练习
Try simplifying each expression before checking the answer.
请先尝试化简每个表达式，再核对答案。
| Question | Answer |
|---|---|
| 7m − 3 + 2m + 8 | 9m + 5 |
| 4x + 2y − x + 3y | 3x + 5y |
| 5a − 2a² + a + a² | 6a − a² |
| 3(2x − 1) + 2(x + 4) | 8x + 5 |
For the last question, expand first: 6x − 3 + 2x + 8, then collect to get 8x + 5.
最后一题要先展开：6x − 3 + 2x + 8，再合并得到 8x + 5。
11. Study Tips | 学习建议
Use different colours or underlining to mark the same variable-power groups before collecting. For 4x + 2y − x + y, mark the x terms in one colour and the y terms in another.
在合并前，用不同颜色或下划线标出相同的变量-指数组合。例如在 4x + 2y − x + y 中，用一种颜色标出 x 项，用另一种颜色标出 y 项。
Work neatly and write the sign before each term as you rearrange. This helps avoid losing negative signs.
书写要整洁，在重新排列时写出每一项前面的符号。这有助于避免漏掉负号。
Practise with expressions involving squares and fractions, since these appear frequently in IGCSE
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