📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, simplifying an expression often begins with collecting like terms. This skill lets you shorten expressions such as 3x + 2y + 5x − y into a neater form before solving equations or substituting values.
在 IGCSE 代数中，化简表达式通常从合并同类项开始。这项技能可以把 3x + 2y + 5x − y 这类表达式整理成更简洁的形式，为解方程或代入求值做准备。
1. What Are Like Terms? | 什么是同类项
Like terms are algebraic terms that have exactly the same variable part, including the same powers. For example, 4x and −7x are like terms because both contain x to the power 1. However, 4x and 4x² are not like terms because the exponents differ.
同类项是指变量部分完全相同的代数项，包括相同的指数。例如，4x 和 −7x 是同类项，因为它们都含有 x 的一次方。但 4x 和 4x² 不是同类项，因为指数不同。
The constants 3 and −8 are also like terms because they have no variable part at all. You can always add or subtract constants directly.
常数 3 和 −8 也是同类项，因为它们不含变量部分。常数可以直接相加或相减。
2. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and look only at the letter part. In 2a, −5a, and 9a, the coefficient changes but the variable a remains the same, so all three are like terms.
要识别同类项，先忽略系数，只看字母部分。在 2a、−5a 和 9a 中，系数不同，但变量 a 相同，所以这三项都是同类项。
xy and x²y are not like terms, since one has x to the power 1 and the other has x squared.
xy 和 3yx 是同类项，因为乘法满足交换律：xy = yx。但 xy 和 x²y 不是同类项，因为前者 x 是一次方，后者 x 是二次方。
3. The Rule for Collecting Like Terms | 合并同类项法则
f like terms, then keep the variable part unchanged. For example, 6x + 3x = (6 + 3)x = 9x.
法则很简单：把同类项的系数相加或相减，变量部分保持不变。例如，6x + 3x = (6 + 3)x = 9x。
This works because the variable part acts as a common factor. Writing 6x + 3x as x(6 + 3) helps show why only the numbers are combined.
这是因为变量部分可以看作公因数。把 6x + 3x 写成 x(6 + 3)，可以更清楚地看出为什么只合并数字部分。
ax + bx = (a + b)x
4. Single Variable Examples | 单变量示例
ms contain m, so combine the coefficients: 7 + 4 − 2 = 9. The simplified expression is 9m.
化简 7m + 4m − 2m。三项都含有 m，因此合并系数：7 + 4 − 2 = 9。化简结果为 9m。
10p − 3p + p, remember that p means 1p. The coefficient sum is 10 − 3 + 1 = 8, giving 8p.
对于 10p − 3p + p，注意 p 表示 1p。系数和为 10 − 3 + 1 = 8，结果为 8p。
Always check the sign in front of each term: 12n − 5n = 7n, while 12n − 15n = −3n.
务必检查每一项前面的符号：12n − 5n = 7n，而 12n − 15n = −3n。
5. Two or More Variables | 两个或多个变量
7b by collecting a terms: 5a − 3a = 2a, and b terms: 2b + 7b = 9b. The answer is 2a + 9b.
当表达式中含有不同字母时，只把变量部分相同的项归为一组。化简 5a + 2b − 3a + 7b：先合并 a 项 5a − 3a = 2a，再合并 b 项 2b + 7b = 9b，结果为 2a + 9b。
It is often helpful to underline or circle each set of like terms with a different mark. This reduces mistakes when several variables appear.
遇到多个变量时，用不同标记划线或圈出每组同类项会很有帮助，能减少错误。
6. Combining Positive and Negative Terms | 正负项合并
same rule. Simplify 4x − 9x + 2x. The calculation is 4 − 9 + 2 = −3, so the answer is −3x.
负项遵循同样的法则。化简 4x − 9x + 2x：计算 4 − 9 + 2 = −3，因此结果为 −3x。
ms give 3y + 8y = 11y, and the constants give −5 − 2 = −7. The simplified form is 11y − 7.
在 3y − 5 + 8y − 2 中，y 项合并为 3y + 8y = 11y，常数项合并为 −5 − 2 = −7。化简结果为 11y − 7。
7. Expressions with Powers | 含幂的表达式
x² + 3x² = 4x². However, x² and x are different and cannot be combined into a single term.
只有变量相同且指数也相同的项才能合并。例如，x² 和 3x² 是同类项，因此 x² + 3x² = 4x²。但 x² 和 x 不同，不能合并为一项。
variable part is a²b in every term, so the coefficient sum is 2 + 5 − 1 = 6, giving 6a²b.
考虑 2a²b + 5a²b − a²b。每一项的变量部分都是 a²b，所以系数和为 2 + 5 − 1 = 6，结果为 6a²b。
x² + x ≠ 2x²
8. Common Mistakes | 常见错误
ncorrect because 3x and 2y do not have the same variable part. The expression 3x + 2y must stay as it is.
常见错误是把不同类项相加，例如把 3x + 2y 写成 5xy。这是不正确的，因为 3x 和 2y 的变量部分不同。3x + 2y 必须保持原样。
ut 2x² + 3x² is not 5x⁴. The exponent never changes when adding or subtracting like terms.
另一个错误是改变指数：2x² + 3x² = 5x² 是正确的，但 2x² + 3x² 不等于 5x⁴。加减同类项时指数不变。
| Incorrect | Correct |
|---|---|
| 3x + 2y = 5xy | 3x + 2y remains 3x + 2y |
| 2x² + 3x² = 5x⁴ | 2x² + 3x² = 5x² |
9. Using Brackets First | 先去括号再合并
+ 6b, then collect a terms: 3a + 4a = 7a, and b terms: 6b − b = 5b. The answer is 7a + 5b.
有些
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