📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, algebraic expressions often look complicated because they contain many similar terms. Collecting like terms is the process of simplifying an expression by adding or subtracting terms that have exactly the same variable part. Mastering this skill is essential before moving on to solving equations, expanding brackets, factorising, and graphing functions.
在IGCSE数学中，代数式常常因为含有许多相似的项而看起来很复杂。合并同类项就是通过将变量部分完全相同的项相加或相减来化简代数式的过程。在继续学习解方程、展开括号、因式分解和函数图像之前，掌握这项技能至关重要。
1. What Are Like Terms? | 什么是同类项
Like terms are terms whose variable parts are exactly the same, including the same letters and the same powers. The numbers in front, called coefficients, can be different. For example, 3x and −5x are like terms because both contain x to the power 1. However, 3x and 3x² are not like terms because the powers of x are different.
同类项是指变量部分完全相同的项，包括相同的字母和相同的次数。前面的数称为系数，系数可以不同。例如，3x 和 −5x 是同类项，因为它们都含有 x 的一次方。但是，3x 和 3x² 不是同类项，因为 x 的次数不同。
3x, −5x, ½x → like terms
4xy, 7xy, −2xy → like terms
2x² and 3x → not like terms
2. Why We Collect Like Terms | 为什么要合并同类项
Collecting like terms makes an expression shorter and easier to use. Instead of writing x + x + x + x, we write 4x. In later topics such as solving linear equations or rearranging formulae, simplifying first reduces errors and shows the structure of the expression clearly.
合并同类项能使表达式更短、更易使用。我们不用写 x + x + x + x，而是写成 4x。在后续学习解一元方程或变换公式时，先化简可以减少错误，并更清楚地展示表达式的结构。
For example, the expression 2x + 3 + 5x − 1 can be written as 7x + 2. This shorter form is much easier to substitute into a formula or solve as part of an equation.
例如，表达式 2x + 3 + 5x − 1 可以写成 7x + 2。这个更短的形式在代入公式或作为方程的一部分求解时要容易得多。
3. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and look only at the variable part. Terms such as 7a, −2a, and 0.5a are like terms. Terms such as 7a, 7b, and 7ab are unlike terms. Constant terms, such as 4 and −9, are also like terms with each other.
识别同类项时，忽略系数，只看变量部分。像 7a、−2a 和 0.5a 这样的项是同类项。像 7a、7b 和 7ab 这样的项不是同类项。常数项（如 4 和 −9）彼此之间也是同类项。
- 5x, 3x, −x → like terms because all contain x
- 4x², 2x → unlike terms because the powers of x differ
- 6xy, −2yx → like terms because xy = yx
- 3, −8, 0.5 → like terms because they are all constants
中文：5x、3x、−x 是同类项，因为都含有 x；4x² 和 2x 不是同类项，因为 x 的次数不同；6xy 和 −2yx 是同类项，因为 xy = yx；3、−8、0.5 是同类项，因为它们都是常数。
4. The Basic Rule: Add or Subtract Coefficients | 基本法则：系数相加减
When collecting like terms, add or subtract the coefficients and keep the variable part unchanged. This rule works because multiplication distributes over addition. For example, 3x means three lots of x, and 5x means five lots of x, so together there are eight lots of x.
合并同类项时，将系数相加或相减，变量部分保持不变。这个法则成立是因为乘法对加法满足分配律。例如，3x 表示三个 x，5x 表示五个 x，所以一共有八个 x。
3x + 5x = (3 + 5)x = 8x
7y − 2y = (7 − 2)y = 5y
4ab + 6ab − ab = (4 + 6 − 1)ab = 9ab
5. Working with Negative Coefficients | 负系数处理
When a term has a negative coefficient, treat the negative sign as part of the coefficient. Use directed number rules: adding a negative is subtraction, and subtracting a negative is addition. This is especially important when an expression contains several negative terms.
当一项的系数为负时，将负号视为系数的一部分。使用有向数规则：加上负数等于减去，减去负数等于加上。这在表达式含有多个负项时尤其重要。
5x − 8x + 3x = (5 − 8 + 3)x = 0x = 0
−2a − 5a = (−2 − 5)a = −7a
4y − (−3y) = 4y + 3y = 7y
6. Combining Terms with Multiple Variables | 含多个变量的项的合并
Terms with more than one variable can be like terms only if every variable and every power match. Treat xy and yx as the same because multiplication is commutative. However, x²y and xy² are not like terms because the powers attached to x and y are different.
含有多个变量的项只有在每个变量和每个次数都相同时才是同类项。因为乘法满足交换律，xy 和 yx 相同。但是，x²y 和 xy² 不是同类项，因为 x 和 y 的次数不同。
2xy + 3yx = 5xy
5a²b − 2a²b = 3a²b
3x²y + 4xy² → cannot be simplified further
7. Simplifying Expressions with Brackets First | 先展开括号再合并
If an expression contains brackets, expand them first, then collect like terms. Be careful with signs when multiplying a bracket by a negative number. Each term inside the bracket must be multiplied by the factor outside.
如果表达式含有括号，需要先展开括号，然后合并同类项。当括号乘以负数时，要注意符号。括号内的每一项都必须乘以外面的因数。
2(x + 3) + 3(x − 1) = 2x + 6 + 3x − 3 = 5x + 3
3(2a − b) − 2(a + b) = 6a − 3b − 2a − 2b = 4a − 5b
In the second example, the negative sign before 2(a + b) changes both a and b into −2a and −2b. This is one of the most common places where marks are lost in IGCSE algebra questions.
在第二个例子中，2(a + b) 前面的负号使 a 和 b 分别变为 −2a 和 −2b。这是IGCSE代数题中最常见的失分点之一。
8. Fraction and Decimal Coefficients | 分数与小数系数
The same rule works for fractional and decimal coefficients. Write all coefficients over a common denominator or convert decimals to fractions when helpful. The variable part still does not change.
同样的法则也适用于分数和小数系数。必要时将所有系数写成同分母分数，或将小数化为分数。变量部分仍然不变。
½x + ¼x = ¾x
0.6x − 0.2x = 0.4x
(2/3)a + (1/6)a = (4/6)a + (1/6)a = (5/6)a
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
A very common mistake is adding unlike terms, such as writing 3x + 2x² as 5x³ or 5x². These are not valid because the variable parts are different. Another mistake is forgetting the invisible 1 in a term like x, which is actually 1x. Students also lose signs when subtracting terms inside brackets.
一个很常见的错误是将不同类项相加，例如把 3x + 2x² 写成 5x³ 或 5x²。这是不正确的，因为变量部分不同。另一个错误是忘记像 x 这样的项中隐藏的系数 1，即 1x。学生在减去括号内的项时还容易丢失符号。
To avoid these errors, underline or circle each set of like terms before combining. Work step by step and double-check the sign of every coefficient.
为了避免这些错误，在合并之前给每组同类项画线或画圈。一步一步计算，并再次检查每个系数的符号。
10. Exam-Style Worked Examples | 考试题型示例
Worked Example 1: Simplify 5a + 3b − 2a + 7b.
示例1：化简 5a + 3b − 2a + 7b。
5a − 2a = 3a, 3b + 7b = 10b → 3a + 10b
Worked Example 2: Simplify 4(2x − 1) − 3(x + 4).
示例2：化简 4(2x − 1) − 3(x + 4)。
8x − 4 − 3x − 12 = 5x − 16
Worked Example 3: Simplify 3x² + 5x − 2x² + x − 7.
示例3：化简 3x² + 5x − 2x² + x − 7。
3x² − 2x² = x², 5x + x = 6x → x² + 6x − 7
Worked Example 4: Simplify 2xy − 3x + 4yx + 5x.
示例4：化简 2xy − 3x + 4yx + 5x。
2xy + 4yx = 6xy, −3x + 5x = 2x → 6xy + 2x
11. Practice Questions and Strategy | 练习题与解题策略
Use a clear strategy for every question: first expand brackets if needed, identify like terms, group them carefully, then add or subtract coefficients. Keep the variable part unchanged.
每道题都使用清晰的策略：如果需要，先展开括号；识别同类项；仔细分组；然后对系数进行加减。保持变量部分不变。
Q1: Simplify 6m + 2n − 4m + 5n.
中文：化简 6m + 2n − 4m + 5n。答案：2m + 7n。
Q2: Simplify 3(2x − 5) + 2(4 − x).
中文：化简 3(2x − 5) + 2(4 − x)。答案：4x − 7。
Q3: Simplify ½x + ¼x.
中文：化简 ½x + ¼x。答案：¾x。
Q4: Simplify 4p² + 3p − 2p² + 6p.
中文：化简 4p² + 3p − 2
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