📚 Collecting Like Terms in IGCSE Mathematics | IGCSE 数学：合并同类项
In IGCSE mathematics, "combining" or "collecting like terms" is one of the most essential algebra skills. It means simplifying an expression by adding or subtracting terms that have exactly the same variable part. For example, 3x + 5x simplifies to 8x, but 3x + 5y cannot be combined into a single term.
在 IGCSE 数学中，“合并”或“合并同类项”是最基本的代数技能之一。它指的是通过加减那些变量部分完全相同的项来化简表达式。例如，3x + 5x 可化简为 8x，但 3x + 5y 不能合并成一项。
1. What Does "Collecting Like Terms" Mean? | 什么是“合并同类项”？
Collecting like terms is the process of simplifying an algebraic expression by combining terms that have identical variable parts. Only the coefficients are added or subtracted; the variable part stays unchanged.
合并同类项是化简代数式的过程，方法是把变量部分完全相同的项合并起来。只有系数进行加减，变量部分保持不变。
3)a = 10a. The variable a is common to both terms, so the coefficients 7 and 3 are added.
例如，7a + 3a = (7 + 3)a = 10a。两个项都含有变量 a，所以系数 7 和 3 相加。
2. Algebraic Terms and Coefficients | 代数项与系数
An algebraic term consists of a number part called the coefficient, a variable part, and possibly an exponent. In the term -4x²y, the coefficient is -4, while the variable part is x²y.
代数项由数字部分（系数）、变量部分以及可能的指数组成。在 -4x²y 中，系数是 -4，变量部分是 x²y。
| Term | Coefficient | Variable part |
|---|---|---|
| 5x | 5 | x |
| -3y² | -3 | y² |
| ½ab | ½ | ab |
| 7 | 7 | none (constant) |
Understanding the coefficient and variable part separately is the key to knowing whether two terms can be combined.
分别理解系数和变量部分是判断两个项能否合并的关键。
3. Like Terms vs Unlike Terms | 同类项与不同类项
Like terms have exactly the same variable part, including the same exponents. Therefore, 2x and -5x are like terms, but 2x and 3x² are not like terms.
同类项的变量部分完全相同，包括相同的指数。因此，2x 和 -5x 是同类项，但 2x 和 3x² 不是同类项。
Similarly, 4ab and -ab are like terms, while 4ab and 4a are not because the variable parts differ. Constants such as 3 and -8 are always like terms with each other.
同样，4ab 和 -ab 是同类项，而 4ab 和 4a 不是，因为变量部分不同。常数项如 3 和 -8 彼此总是同类项。
Like: 3x and -7x, 5y² and 2y², 4ab and -ab | Unlike: 2x and 3x², 5ab and 5a, 3y and 3
4. The Golden Rule of Combining | 合并的黄金法则
You can only combine terms when their variable parts match exactly. To combine, add or subtract the coefficients and keep the variable part unchanged.
只有当两个项的变量部分完全相同时，才能合并。合并时，系数相加或相减，变量部分保持不变。
px + qx = (p + q)x
For example, 9m - 4m = (9 - 4)m = 5m. The variable m remains as m, not m².
例如，9m - 4m = (9 - 4)m = 5m。变量 m 仍然是 m，而不是 m²。
5. Combining Positive and Negative Coefficients | 合并正负系数
efficient. For example, 6k - 10k + 3k means (+6) + (-10) + (+3) = -1, so the result is -k.
符号是常见的错误来源。把每项前面的符号当作系数的一部分。例如，6k - 10k + 3k 表示 (+6) + (-10) + (+3) = -1，因此结果是 -k。
p + 5p - 2p = (-8 + 5 - 2)p = -5p. Always work from left to right with the signs attached.
另一个例子：-8p + 5p - 2p = (-8 + 5 - 2)p = -5p。计算时始终从左到右，并保留符号。
6. Working with Powers and Multiple Variables | 处理幂与多变量
When powers appear, terms are like only if the variable and its exponent match. For example, x² and x are not like terms. Similarly, xy and x²y are not like terms.
当出现幂时，只有变量及其指数都相同才是同类项。例如，x² 和 x 不是同类项。同样，xy 和 x²y 也不是同类项。
Combine separately: 5x² + 2x - 3x² - x = (5 - 3)x² + (2 - 1)x = 2x² + x.
分别合并：5x² + 2x - 3x² - x = (5 - 3)x² + (2 - 1)x = 2x² + x。
5xy = 8xy, but 3xy + 5x cannot be combined because one term has y and the other does not.
对于多个变量，3xy + 5xy = 8xy，但 3xy + 5x 不能合并，因为一个项含有 y，另一个没有。
7. Removing Brackets Before Combining | 先去括号再合并
if necessary, then collect like terms. For instance, 3(x + 2) - 2x = 3x + 6 - 2x = x + 6.
如果表达式含有括号，必要时先展开，再合并同类项。例如，3(x + 2) - 2x = 3x + 6 - 2x = x + 6。
1) = 4 - x - 1 = 3 - x. The minus sign changes the signs of all terms inside the bracket.
括号前有负号时要小心：4 - (x + 1) = 4 - x - 1 = 3 - x。负号会改变括号内所有项的符号。
Another example: 5 - 2(3x - 1) = 5 - 6x + 2 = 7 - 6x.
另一个例子：5 - 2(3x - 1) = 5 - 6x + 2 = 7 - 6x。
8. Fraction and Decimal Coefficients | 分数与小数系数
The same rules apply with fractions and decimals. For fraction coefficients, find a common denominator when adding or subtracting the coefficients.
分数和小数遵循相同的规则。对于分数系数，加减系数时要先通分。
Example: (3/4)x + (1/2)x = (3/4 + 2/4)x = (5/4)x = 1¼ x.
例如，(3/4)x + (1/2)x = (3/4 + 2/4)x = (5/4)x = 1¼ x。
Always keep the variable part unchanged even when coefficients are fractions or decimals.
对于小数，0.3y + 1.7y = 2.0y = 2y。即使系数是分数或小数，变量部分也始终保持不变。
9. Real-Life Applications: Perimeter and Cost | 实际应用：周长与成本
+ 1 and width 2x - 4, its perimeter is 2(3x + 1) + 2(2x - 4) = 6x + 2 + 4x - 8 = 10x - 6.
合并同类项经常出现在周长问题中。如果矩形的长为 3x + 1，宽为 2x - 4，则周长为 2(3x + 1) + 2(2x - 4) = 6x + 2 + 4x - 8 = 10x - 6。
It is also used in cost problems: if apples cost x dollars and bananas cost y dollars, the total cost of 3 apples and 5 bananas is 3x + 5y, which cannot be simplified further because the terms are not like.
也用于成本问题：如果苹果每个 x 元，香蕉每个 y 元，则 3 个苹果和 5 个香蕉的总价为 3x + 5y，由于不是同类项，不能再合并。
10. Common Mistakes and How to Avoid Them | 常见错误与避免方法
for example writing 2x + 3x = 5x². This is incorrect; the variable part x does not change.
常见错误是系数相加后还把变量部分相乘，例如写成 2x + 3x = 5x²。这是错误的；变量部分 x 不变。
- Do not combine unlike terms: 2x + 3x² cannot be simplified to a single term.
- 不合并不同类项：2x + 3x² 不能合并成一项。
- Keep the sign with each term when rearranging: 5 - 2x + 3 = 8 - 2x, not 8 + 2x.
- 重新排列时保留每项的符号：5 - 2x + 3 = 8 - 2x，而不是 8 + 2x。
- When expanding brackets, apply the multiplier to every term inside, including signs.
- 展开括号时，将乘数作用于括号内的每一项，包括符号。
11. Worked Examples from IGCSE Style Questions | IGCSE 类型例题
Simplify 5a - 3b + 2a + 7b. Group like terms: 5a + 2a - 3b + 7b = 7a + 4b.
化简 5a - 3b + 2a + 7b。合并同类项：5a + 2a - 3b + 7b = 7a + 4b。
implify 4x² - x + 2x² + 3x - 5. Collect like terms: 4x² + 2x² - x + 3x - 5 = 6x² + 2x - 5.
化简 4x² - x + 2x² + 3x - 5。合并同类项：4x² + 2x² - x + 3x - 5 = 6x² + 2x - 5。
Expand and simplify 2(x + 3) - (x - 4). First expand: 2x + 6 - x + 4 = x + 10.
展开并化简 2(x + 3) - (x - 4)。先展开：2x + 6 - x + 4 = x + 10。
12. Practice Tips and Summary | 练习提示与总结
To master collecting like terms, first circle or underline each like term with the same symbol, then rearrange the expression so like terms are together. Check that the variable part has not changed in your final answer.
要掌握合并同类项，首先可以用相同记号圈出或划出每类同类项，然后重新排列表达式，使同类项相邻。检查最终答案中变量部分是否保持不变。
Key points: like terms have identical variable parts; add or subtract coefficients only; signs travel with terms; expand brackets before collecting.
要点：同类项的变量部分完全相同；只加减系数；符号随项移动；先展开括号再合并。
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