📚 Collecting Like Terms in IGCSE Mathematics | IGCSE 数学：合并同类项
In IGCSE Mathematics, simplifying algebraic expressions often starts with collecting like terms. A 'term' is a single number, variable, or product of numbers and variables, such as 5, x, 3x², or -2xy. Like terms have exactly the same variable parts and exponents, so they can be added or subtracted. This skill is tested in algebra, equation solving, and geometry problems.
在 IGCSE 数学中，化简代数式通常从合并同类项开始。'项' 是一个单独的数、变量或数与变量的乘积，例如 5、x、3x² 或 -2xy。同类项具有完全相同的变量部分和指数，因此可以相加或相减。这一技能在代数、方程求解和几何题中都经常考查。
1. What Are Like Terms? | 什么是同类项？
A term is made of a coefficient and a variable part. For example, in 7x², the coefficient is 7 and the variable part is x².
项由系数和变量部分组成。例如，在 7x² 中，系数是 7，变量部分是 x²。
Two or more terms are 'like' if their variable parts are identical, including powers. So 4x and -11x are like terms, but 4x and 4x² are not.
两个或多个项如果变量部分完全相同（包括指数），就称为同类项。因此 4x 和 -11x 是同类项，但 4x 和 4x² 不是。
| Term 1 | Term 2 | Like? | Reason |
|---|---|---|---|
| 5x | -3x | Yes | Both have x¹ |
| 2x² | 9x² | Yes | Both have x² |
| 4xy | -xy | Yes | Both have xy |
| 7x | 7x² | No | Powers differ |
| 6a | 6b | No | Different variables |
| 3x²y | 5xy² | No | Exponents on x and y differ |
2. The Golden Rule of Collecting | 合并的核心规则
You can only add or subtract coefficients of like terms. The variable part stays unchanged.
你只能对同类项的系数进行加减，变量部分保持不变。
3x + 5x = (3 + 5)x = 8x
Here, the coefficients 3 and 5 are added, and x remains unchanged.
这里将系数 3 和 5 相加，x 保持不变。
3. Identifying Like Terms in Larger Expressions | 在复杂表达式中识别同类项
Before collecting, scan the expression and group terms that share the same variable part. Circle or underline them if needed.
合并之前，先扫描表达式，将具有相同变量部分的项分组。如果需要，可以圈出或划线标记。
Example: 2x + 3y - 5x + y + 4. The x terms are 2x and -5x, the y terms are 3y and y, and 4 is a constant.
示例：2x + 3y - 5x + y + 4。x 项是 2x 和 -5x，y 项是 3y 和 y，4 是常数项。
Collect them: (2x - 5x) + (3y + y) + 4 = -3x + 4y + 4.
合并得到：(2x - 5x) + (3y + y) + 4 = -3x + 4y + 4。
4. Handling Negative Signs | 处理负号
A term's sign belongs to its coefficient. When moving terms around, take the sign in front of the term with it.
项的符号属于其系数。移动项时，要把项前面的符号一起带走。
Simplify 8a - 3b + 2a - 5b. Group as (8a + 2a) + (-3b - 5b) = 10a - 8b.
化简 8a - 3b + 2a - 5b。分组为 (8a + 2a) + (-3b - 5b) = 10a - 8b。
subtracting a term is the same as adding its negative. For example, 6x - 2x = 4x, not 8x.
注意：减去一个项等于加上它的相反数。例如 6x - 2x = 4x，而不是 8x。
5. Constants and Different Powers | 常数项与不同次幂
Constants combine only with constants. x² and x look similar but are NOT like terms because their powers differ.
常数项只能与常数项合并。x² 和 x 看起来相似但不是同类项，因为它们的指数不同。
² terms: x² - 2x² = -x². Combine x terms: 3x + 5x = 8x. Constant: 7. Result: -x² + 8x + 7.
化简 x² + 3x - 2x² + 5x + 7。合并 x² 项：x² - 2x² = -x²。合并 x 项：3x + 5x = 8x。常数项为 7。结果为 -x² + 8x + 7。
6. Expanding Then Collecting | 先展开再合并
Many IGCSE questions require you to expand brackets first, then collect like terms. For example, simplify 3(x + 2) + 2(x - 5).
许多 IGCSE 题目要求先展开括号，再合并同类项。例如，化简 3(x + 2) + 2(x - 5)。
+ 2x - 10. Collect x terms: 3x + 2x = 5x. Collect constants: 6 - 10 = -4. Answer: 5x - 4.
展开：3x + 6 + 2x - 10。合并 x 项：3x + 2x = 5x。合并常数项：6 - 10 = -4。答案：5x - 4。
ry term inside by the negative. Example: 4(2x - 1) - 2(x + 3) = 8x - 4 - 2x - 6 = 6x - 10.
如果括号外是负号，记得将括号内每一项都乘以该负号。示例：4(2x - 1) - 2(x + 3) = 8x - 4 - 2x - 6 = 6x - 10。
7. Worked Example with Multiple Variables | 多变量综合示例
Simplify 5xy - 3x² + 2xy + x² - 4.
化简 5xy - 3x² + 2xy + x² - 4。
5xy + 2xy = 7xy; x² terms: -3x² + x² = -2x²; constant: -4. So the answer is 7xy - 2x² - 4.
分组同类项：xy 项：5xy + 2xy = 7xy；x² 项：-3x² + x² = -2x²；常数项：-4。因此答案是 7xy - 2x² - 4。
It is common to write terms in any order, but examiners often prefer descending powers. Here the standard form is -2x² + 7xy - 4.
项的顺序可以任意，但考官通常偏好按降幂排列。这里标准形式为 -2x² + 7xy - 4。
8. Using Collecting Like Terms in Geometry | 在几何中应用合并同类项
IGCSE questions may ask you to find a perimeter or sum of lengths and simplify it. For example, a triangle has sides 2x + 3, 4x - 1, and 5x + 2. Find the perimeter.
IGCSE 题目可能会要求你求周长或长度之和并化简。例如，一个三角形的三条边分别为 2x + 3、4x - 1 和 5x + 2，求周长。
ollect x terms: 2x + 4x + 5x = 11x. Collect constants: 3 - 1 + 2 = 4. Perimeter = 11x + 4.
周长 = (2x + 3) + (4x - 1) + (5x + 2)。合并 x 项：2x + 4x + 5x = 11x。合并常数项：3 - 1 + 2 = 4。周长 = 11x + 4。
This shows that collecting like terms is not just abstract algebra; it helps simplify real measurement expressions.
这表明合并同类项不仅是抽象代数，它还能化简实际的度量表达式。
9. Common Mistakes to Avoid | 常见错误与避免方法
Mistake 1: Adding unlike terms. Example: writing 3x + 4x² as 7x². This is wrong because x and x² are not like terms.
错误 1：将非同类项相加。例如，把 3x + 4x² 写成 7x²。这是错误的，因为 x 和 x² 不是同类项。
Mistake 2: Losing the sign. In 5 - 2x + 3x, the terms -2x and +3x combine to +x, so the answer is 5 + x, not 5 - 5x.
错误 2：丢失符号。在 5 - 2x + 3x 中，-2x 和 +3x 合并为 +x，所以答案是 5 + x，而不是 5 - 5x。
Mistake 3: Forgetting to distribute a negative to all terms in a bracket. Write intermediate steps to avoid this.
错误 3：忘记将负号分配给括号内的所有项。写出中间步骤可以避免这种错误。
10. Quick Practice and Mental Check | 快速练习与心算检验
Try simplifying 6p - 2q + 3p + q - 5. The p terms make 9p, the q terms make -q, and the constant is -5,
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