📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, collecting like terms is the first essential skill for simplifying expressions. A term is a single number, a variable, or a product of numbers and variables, and like terms have exactly the same variable part.
在 IGCSE 代数中，合并同类项是化简表达式的第一个基本技能。项是单个数字、变量或数字与变量的乘积；同类项具有完全相同的变量部分。
1. What Are Like Terms? | 什么是同类项
In an expression such as 3x + 5y - 2x + 7, each part separated by + or - is a term. The terms 3x and -2x are like terms because they both contain the variable x to the same power. The number in front is called the coefficient, and it does not change whether terms are like.
在表达式 3x + 5y - 2x + 7 中，每个由 + 或 - 分隔的部分都是一个项。3x 和 -2x 是同类项，因为它们都含有相同幂次的变量 x。前面的数字称为系数，它不会改变两项是否为同类项。
- Like terms: 3x, -2x, 0.5x | 同类项：3x、-2x、0.5x
- Not like terms: 3x and 3x² | 不是同类项：3x 与 3x²
- Constant terms: 7, -4, 12 are like terms | 常数项：7、-4、12 是同类项
2. Why We Collect Like Terms | 为什么要合并同类项
Collecting like terms makes an expression shorter and clearer. It is also required before solving equations, drawing graphs, or substituting values. Without simplifying first, later algebra becomes harder and more prone to mistakes.
合并同类项可以让表达式更短、更清晰。在解方程、画图像或代入求值之前，通常也需要先合并同类项。如果不先化简，后续的代数运算会更困难，也更容易出错。
3. Identifying Like Terms | 识别同类项
Two terms are like only when their variable letters and powers are exactly the same. Constants are like terms with each other. The coefficient can be different: 4x and -7x are like, but 4x and 4y are not.
只有当两个项的变量字母和幂次完全相同时，它们才是同类项。常数项彼此是同类项。系数可以不同：4x 和 -7x 是同类项，但 4x 和 4y 不是。
| Terms | Like? | Reason |
| 5a and -2a | Yes | Same variable a |
| 4xy and 3x | No | Different variable parts |
| 2x² and 5x² | Yes | Same variable and same power |
4. Adding and Subtracting Like Terms | 加减同类项
To add or subtract like terms, add or subtract the coefficients only and keep the variable part unchanged. For example, 3x + 2x means three x values plus two x values, giving five x values.
加减同类项时，只加减系数，变量部分保持不变。例如，3x + 2x 表示三个 x 加上两个 x，得到五个 x。
3x + 2x = (3 + 2)x = 5x
Subtraction works the same way: 5y - 2y gives 3y. The variable y does not change; only the numbers 5 and 2 are subtracted.
减法同理：5y - 2y 得到 3y。变量 y 不变，只对数字 5 和 2 做减法。
5y - 2y = (5 - 2)y = 3y
5. Combining More Than Two Terms | 合并多个同类项
When an expression has several groups of like terms, collect each group separately. It helps to underline or circle matching terms before adding them together.
当一个表达式中有多组同类项时，应分别合并每一组。可以先在同类项下面划线或画圈，然后再相加。
4a + 3b - 2a + b = (4a - 2a) + (3b + b) = 2a + 4b
Here the a terms are 4a and -2a, which combine to 2a. The b terms are 3b and b, which combine to 4b.
这里 a 项是 4a 和 -2a，合并为 2a。b 项是 3b 和 b，合并为 4b。
6. Different Variables and Powers | 不同变量与幂次
Terms with different variables cannot be combined. For example, 2x + 3y stays as 2x + 3y because x and y are different. Also, x and x² are not like terms because the powers are different.
不同变量的项不能合并。例如，2x + 3y 仍写作 2x + 3y，因为 x 和 y 不同。同样，x 和 x² 也不是同类项，因为幂次不同。
4x² + 2x² = 6x², but 4x² + 2x cannot be combined
The power must match exactly. Only terms with the same variable and the same exponent are like terms.
幂次必须完全一致。只有变量相同且指数相同的项才是同类项。
7. Expressions with Brackets | 含有括号的表达式
If an expression contains brackets, expand the brackets first, then collect like terms. Multiply each term inside the bracket by the number or sign outside.
如果表达式中含有括号，应先去括号，再合并同类项。将括号内的每一项都乘以外面的数或符号。
2(x + 3) + 3(x - 1) = 2x + 6 + 3x - 3 = 5x + 3
n the expression -(x - 4), the negative sign changes every sign inside: -(x - 4) = -x + 4.
括号前有负号时要特别注意。在 -(x - 4) 中，负号会改变括号内每一项的符号：-(x - 4) = -x + 4。
8. Common Mistakes | 常见错误
Students often combine terms that are not alike. Remember that addition and subtraction cannot change variables or powers: the variable part must stay the same.
学生常常把不是同类项的项合并在一起。记住，加减法不能改变变量或幂次：变量部分必须保持不变。
- Mistake 1: 3x + 2y = 5xy is wrong because x and y are different. | 错误一：3x + 2y = 5xy 是错误的，因为 x 和 y 不同。
- Mistake 2: x + x = x² is wrong; the correct answer is 2x. | 错误二：x + x = x² 是错误的；正确答案是 2x。
- Mistake 3: written as 3x - 3, but the -2x must be subtracted from 5x, so 5x - 2x = 3x, giving 3x - 3. | 错误三：5x - 3 - 2x 常被误写，但 -2x 应从 5x 中减去，所以 5x - 2x = 3x，得到 3x - 3。
9. Worked Examples | 典型例题
Example 1: Simplify 5x - 3y + 2x + y - 4. First group the x terms, then the y terms, then the constant.
例 1：化简 5x - 3y + 2x + y - 4。先合并 x 项，再合并 y 项，最后处理常数项。
5x + 2x = 7x, -3y + y = -2y, so the answer is 7x - 2y - 4
Example 2: Simplify 4a² - 3a + 2a² + 5a. The a² terms and the a terms must be collected separately because the powers are different.
例 2：化简 4a² - 3a + 2a² + 5a。a² 项和 a 项必须分别合并，因为幂次不同。
4a² + 2a² = 6a², -3a + 5a = 2a, so the answer is 6a² + 2a
Example 3: Expand and simplify 2(3x + 1) - (x - 4). Expand first, then combine the x terms and constants.
例 3：先去括号再化简 2(3x + 1) - (x - 4)。先展开，再合并 x 项和常数项。
2(3x + 1) - (x - 4) = 6x + 2 - x + 4 = 5x + 6
10. Exam-Style Practice | 考试型练习
Try these IGCSE-style questions. Simplify each expression fully and check that the variable parts are correct in every term.
尝试以下 IGCSE 型练习题。完全化简每个表达式，并检查每一项的变量部分是否正确。
- Simplify 6p - 2q + 3p + q. | 化简 6p - 2q + 3p + q。 Answer: 9p - q
- Simplify 4x² + 3x - 2x² + x - 5. | 化简 4x² + 3x - 2x² + x - 5。 Answer: 2x² + 4x - 5
- Expand and simplify 3(m + 2) + 2(2m - 1). | 去括号并化简 3(m + 2) + 2(2m - 1)。 Answer: 7m + 4
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