📚 Collecting Like Terms | 合并同类项
In algebra, an expression such as 3x + 5x can be made much simpler. This process is called collecting like terms, and it is one of the most important skills in IGCSE Mathematics.
在代数中，像 3x + 5x 这样的式子可以化简很多。这个过程叫做合并同类项，是 IGCSE 数学中最重要的技能之一。
1. Algebra as a Language | 代数语言
Terms are the building blocks of algebraic expressions. In 3x + 5x − 2, the terms are 3x, 5x and −2.
项是代数式的基本组成部分。在 3x + 5x − 2 中，项分别是 3x、5x 和 −2。
A term has a sign, a coefficient and a variable part. For example, in −4x², the coefficient is −4 and the variable part is x².
一个项由符号、系数和变量部分组成。例如，在 −4x² 中，系数是 −4，变量部分是 x²。
Combining terms correctly is like combining the same type of object: you can add 3 apples and 5 apples, but not apples and oranges.
正确合并项就像合并同一种物品：你可以把 3 个苹果和 5 个苹果加在一起，但不能把苹果和橙子加在一起。
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.
同类项是指变量部分完全相同（包括相同的幂次）的项。系数可以不同。
Example: 3x and −7x are like terms because both contain x. However, 3x² and 3x are not like terms because the powers of x are different.
例如：3x 和 −7x 是同类项，因为它们都含有 x。但是 3x² 和 3x 不是同类项，因为 x 的幂次不同。
Constants such as 5, −2 and 12 are also like terms, because they have no variable part.
常数项如 5、−2 和 12 也是同类项，因为它们没有变量部分。
| Like terms / 同类项 | Not like terms / 不是同类项 |
| 5x and −3x | 5x and 5x² |
| 4y² and 2y² | 4y and 4y² |
| 7 and −9 | 7 and 7x |
3. Identifying Like Terms Quickly | 快速识别同类项
When scanning an expression, ignore the coefficient at first and focus on the letters and their exponents.
快速扫描一个式子时，先忽略系数，重点关注字母及其指数。
For 2xy, −5xy and xy, all three are like terms because the variable part is xy.
对于 2xy、−5xy 和 xy，这三项都是同类项，因为变量部分都是 xy。
Remember that xy and yx are usually treated as the same variable part because multiplication is commutative.
请记住，xy 和 yx 通常被视为相同的变量部分，因为乘法满足交换律。
4. The Rule for Collecting Like Terms | 合并同类项的规则
To collect like terms, add or subtract the coefficients only. Keep the variable part exactly the same.
合并同类项时，只对系数进行加法或减法运算。变量部分保持不变。
This works because 3x + 5x means adding 3 lots of x and 5 lots of x, giving 8 lots of x.
这是因为 3x + 5x 表示 3 个 x 加 5 个 x，得到 8 个 x。
3x + 5x = (3 + 5)x = 8x
In general, if two terms have the same variable part, add or subtract their coefficients.
一般来说，如果两项具有相同的变量部分，就对它们的系数进行加法或减法。
ax + bx = (a + b)x
5. Worked Example: One Variable | 单变量例题
Simplify 7x − 2x + 3x.
化简 7x − 2x + 3x。
All three terms are like terms, so add the coefficients: 7 − 2 + 3 = 8.
三项都是同类项，因此将系数相加：7 − 2 + 3 = 8。
7x − 2x + 3x = 8x
Simplify 9y + 4 − 3y − 10.
化简 9y + 4 − 3y − 10。
Collect the y terms: 9y − 3y = 6y. Collect the constants: 4 − 10 = −6.
合并 y 项：9y − 3y = 6y。合并常数项：4 − 10 = −6。
9y + 4 − 3y − 10 = 6y − 6
6. Worked Example: Two Variables | 双变量例题
Simplify 4a + 3b − a + 2b.
化简 4a + 3b − a + 2b。
First group the a terms: 4a − a = 3a.
首先合并 a 项：4a − a = 3a。
Then group the b terms: 3b + 2b = 5b.
然后合并 b 项：3b + 2b = 5b。
4a + 3b − a + 2b = 3a + 5b
Simplify 6x − 2y + 3x + 5y − 4.
化简 6x − 2y + 3x + 5y − 4。
Combine x terms: 6x + 3x = 9x. Combine y terms: −2y + 5y = 3y. The constant is −4.
合并 x 项：6x + 3x = 9x。合并 y 项：−2y + 5y = 3y。常数项为 −4。
6x − 2y + 3x + 5y − 4 = 9x + 3y − 4
7. Dealing with Powers | 处理幂次
x, x² and x³ are completely different terms and cannot be combined with each other.
x、x² 和 x³ 是完全不同的项，不能相互合并。
Simplify 2x² + 5x − x² + 3x.
化简 2x² + 5x − x² + 3x。
Combine the x² terms: 2x² − x² = x². Combine the x terms: 5x + 3x = 8x.
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图