📚 IGCSE Maths: Collecting Like Terms | IGCSE 数学：合并同类项
Collecting like terms is one of the most important early algebra skills in IGCSE Mathematics. It is used whenever you simplify an expression, solve an equation, expand brackets, or rearrange a formula. This revision guide explains the rules, worked examples, and common pitfalls step by step.
合并同类项是 IGCSE 数学中最重要的早期代数技能之一。每当你要化简表达式、解方程、展开括号或变形公式时，都会用到它。本复习指南将逐步讲解规则、例题和常见误区。
1. What Are Like Terms? | 什么是同类项
A term is a number, a variable, or a product of numbers and variables. Like terms have exactly the same variable part, including the same powers. For example, 4x and -7x are like terms, while 4x and 4x² are not.
项可以是一个数、一个变量，或数与变量的乘积。同类项具有完全相同的变量部分，包括相同的幂。例如，4x 与 -7x 是同类项，而 4x 与 4x² 不是。
Examples of like terms: 5y, -2y and 0.5y. Examples of unlike terms: 3x, 3x² and 3y. Only the coefficient may be different; the variable structure must be identical.
同类项示例：5y、-2y 和 0.5y。不同类项示例：3x、3x² 和 3y。只有系数可以不同，变量结构必须完全相同。
2. The Golden Rule: Only Combine Like Terms | 黄金法则：只能合并同类项
You can only add or subtract terms that are like. The coefficients are combined numerically, while the variable part stays unchanged. For example, 3a + 5a means (3 + 5)a, which equals 8a.
只能对同类项进行加减。系数在数值上相加，变量部分保持不变。例如，3a + 5a 表示 (3 + 5)a，结果为 8a。
3a + 5a = (3 + 5)a = 8a
If the variable parts are different, the expression cannot be simplified by addition or subtraction. For example, 3a + 5b must remain as 3a + 5b.
如果变量部分不同，则不能通过加减进行化简。例如，3a + 5b 必须保留为 3a + 5b。
3. Single-Variable Examples | 单变量示例
Simplify 7x − 2x + 11x. Since all three terms have the same variable x, combine their coefficients: 7 − 2 + 11 = 16. The result is 16x.
化简 7x − 2x + 11x。因为三项都含有相同的变量 x，所以合并系数：7 − 2 + 11 = 16，结果为 16x。
7x − 2x + 11x = 16x
In the expression 9m + 4 − 3 + m, first group the m terms and the constant terms. Combine 9m + m = 10m and 4 − 3 = 1, giving 10m + 1.
在表达式 9m + 4 − 3 + m 中，先将含 m 的项与常数项分别分组。合并 9m + m = 10m，4 − 3 = 1，得到 10m + 1。
4. Multi-Variable Terms | 多变量项
Terms such as xy and yx are like terms because multiplication is commutative; the order of variables does not matter. However, xy and x²y are not like terms because the power of x is different.
xy 与 yx 是同类项，因为乘法满足交换律，变量顺序不影响。但 xy 与 x²y 不是同类项，因为 x 的幂不同。
Simplify 3xy + 5yx − 2x²y. Combine 3xy + 5yx = 8xy, then the −2x²y term remains separate because it has a different power of x. The final answer is 8xy − 2x²y.
化简 3xy + 5yx − 2x²y。合并 3xy + 5yx = 8xy，而 −2x²y 因 x 的幂不同需要保留。最终答案为 8xy − 2x²y。
5. Powers and Roots | 幂与根
Only terms with identical powers are like terms. For example, x and x² are unlike, and √x and x are unlike. This is a very common IGCSE error point.
只有幂完全相同的项才是同类项。例如，x 与 x² 不是同类项，√x 与 x 也不是。这是 IGCSE 中非常常见的易错点。
Simplify 2x² + 3x + 5x² − x. Group the squared terms together and the linear terms together: 2x² + 5x² = 7x², and 3x − x = 2x. The answer is 7x² + 2x.
化简 2x² + 3x + 5x² − x。将平方项和一次项分别分组：2x² + 5x² = 7x²，3x − x = 2x。答案为 7x² + 2x。
2x² + 3x + 5x² − x = 7x² + 2x
6. Negative Coefficients and Subtraction | 负系数与减法
Treat the sign in front of a term as part of its coefficient. For 5p − 8p + 2p, combine the coefficients as 5 − 8 + 2 = −1, so the result is −p.
把项前面的符号看作系数的一部分。对于 5p − 8p + 2p，合并系数 5 − 8 + 2 = −1，结果为 −p。
5p − 8p + 2p = −1p = −p
In 4k − 3 − 2k + 7, collect the like terms: 4k − 2k = 2k and −3 + 7 = 4. The simplified form is 2k + 4.
在 4k − 3 − 2k + 7 中，合并同类项：4k − 2k = 2k，−3 + 7 = 4。化简结果为 2k + 4。
7. Fractions and Decimals | 分数与小数
The same collecting rule applies when coefficients are fractions or decimals. Simplify ½x + ¾x by adding the coefficients: ½ + ¾ = 2/4 + 3/4 = 5/4, so the answer is 5/4 x or 1¼x.
当系数为分数或小数时，同样的合并规则也适用。化简 ½x + ¾x，将系数相加：½ + ¾ = 2/4 + 3/4 = 5/4，因此答案为 5/4 x 或 1¼x。
½x + ¾x = 1¼x
For decimals, 0.2t + 1.8t = 2.0t, which is simply written as 2t. Keep the variable part unchanged after adding the decimal coefficients.
对于小数，0.2t + 1.8t = 2.0t，通常直接写作 2t。小数系数相加后，变量部分保持不变。
8. Using a Table to Recognise Like Terms | 用表格识别同类项
A quick check can prevent unnecessary mistakes: do the terms have the same variables and the same exponents? If both answers are yes, they are like terms. Use this table to test yourself.
快速检查可以避免不必要的错误：这两项是否具有相同的变量和相同的指数？如果两个条件都满足，它们就是同类项。你可以用下面的表格自测。
| Pair | Like? | Reason |
|---|---|---|
| 6x and 2x | Yes | Same x¹ |
| 4a² and 3a | No | Different powers |
| 5xy and -2yx | Yes | Same variables |
| 7 and -3 | Yes | Both constants |
Remember that constants are always like terms with other constants, but they are never like terms with variables such as x or y.
请记住，常数与其他常数始终是同类项，但常数与 x、y 等变量永远不是同类项。
9. Common Mistakes to Avoid | 常见错误
Never combine unlike terms. A typical error is writing 2x + 3y as 5xy, but this is incorrect because x and y are different variables. The expression 2x + 3y cannot be simplified further.
切勿合并不同类项。一个典型错误是把 2x + 3y 写成 5xy，但这是错误的，因为 x 和 y 是不同的变量。表达式 2x + 3y 不能进一步化简。
Another common mistake is changing powers. For example, x + x does not equal x². Since both terms are x¹, they combine to 2x. Also, 4x + 2 cannot simplify to 6x because 2 is a constant, not an x term.
另一个常见错误是改变幂。例如，x + x 不等于 x²。因为两项都是 x¹，它们合并为 2x。此外，4x + 2 不能化简为 6x，因为 2 是常数，不是含 x 的项。
x + x = 2x, not x²
10. Simplifying Before Substitution | 先化简再代入
In IGCSE questions, you often need to substitute values into an expression. Simplifying by collecting like terms first makes substitution faster and reduces arithmetic errors.
在 IGCSE 题目中，经常需要将数值代入表达式。先通过合并同类项进行化简，可以让代入更快，并减少计算错误。
For example, if x = 2, evaluate 3x + 5x − 2x. Instead of calculating each term separately, simplify first: 3x + 5x − 2x = 6x, then substitute x = 2 to get 12.
例如，当 x = 2 时，求 3x + 5x − 2x 的值。可以先化简：3x + 5x − 2x = 6x，然后代入 x = 2 得到 12。
3x + 5x − 2x = 6x; 6 × 2 = 12
11. Connecting to Expanding and Solving | 与展开、解方程的联系
After expanding brackets, you should always collect like terms to fully simplify. For example, expand 2(x + 3) + 3(x − 1) to get 2x + 6 + 3x − 3, then collect like terms: 5x + 3.
展开括号后，一定要合并同类项才能完全化简。例如，展开 2(x + 3) + 3(x − 1) 得到 2x + 6 + 3x − 3，然后合并同类项：5x + 3。
2(x + 3) + 3(x − 1) = 5x + 3
When solving equations, simplify each side by collecting like terms before using inverse operations. This makes the steps clearer and reduces the chance of sign errors.
解方程时，先通过合并同类项化简等号两边，再使用逆运算。这会让步骤更清晰，并减少符号错误的可能性。
12. Exam-Style Practice and Answers | 考试风格练习与答案
Apply the rules by simplifying these expressions before checking the answers: (a) 8x − 3x + 5x, (b) 2a + 3b + 4a − b, (c) 5p² + p − 2p² + 4p, (d) 3xy + 5x + 2yx − x.
在查看答案前，先应用规则化简下列各式：(a) 8x − 3x + 5x，(b) 2a + 3b + 4a − b，(c) 5p² + p − 2p² + 4p，(d) 3xy + 5x + 2yx − x。
Answers: (a) 10x, (b) 6a + 2b, (c) 3p² + 5p, (d) 5xy + 4x. Check each step to confirm you grouped only like terms and kept the signs correct.
答案：(a) 10x，(b) 6a + 2b，(c) 3p² + 5p，(d) 5xy + 4x。请逐步检查，确认你只合并了同类项，并且符号正确。
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