📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, simplifying algebraic expressions is a core skill. The first major step is collecting like terms. This means grouping terms that have exactly the same variable part and then adding or subtracting their coefficients. This article explains how to identify like terms, combine them correctly, and avoid common errors.
在 IGCSE 数学中，化简代数式是一项核心技能。第一步重要操作就是合并同类项。这意味着把变量部分完全相同的项归为一组，然后加减它们的系数。本文将讲解如何识别同类项、正确合并它们，并避开常见错误。
1. What Are Algebraic Terms? | 什么是代数项？
An algebraic term is a number, a variable, or a product of numbers and variables. For example, 5x, -3y, 7, and 2ab are all terms. In a term, the number part is called the coefficient, and the letter part is called the variable part.
代数项是一个数、一个变量，或数与变量的乘积。例如 5x、-3y、7 和 2ab 都是项。在一个项中，数字部分称为系数，字母部分称为变量部分。
Terms are usually separated by + or - signs. In the expression 4x + 3y - 2, the three terms are 4x, 3y, and -2. Notice that the sign belongs to the term that follows it.
项通常由加号或减号分隔。在表达式 4x + 3y - 2 中，三个项分别是 4x、3y 和 -2。注意符号属于它后面的项。
2. Like Terms: The Key Idea | 同类项：核心概念
Two or more terms are called like terms if their variable parts are exactly the same. That means the same letters and the same powers. For example, 3x and -7x are like terms because both contain x to the power 1.
如果两个或多个项的变量部分完全相同，它们就称为同类项。也就是说，相同的字母和相同的指数。例如 3x 和 -7x 是同类项，因为两者都含有 x 的一次方。
In contrast, 3x and 3x² are not like terms. The variable part is different: one is x, and the other is x². Only identical variable parts may be combined by addition or subtraction.
相比之下，3x 和 3x² 不是同类项。变量部分不同：一个是 x，另一个是 x²。只有完全相同的变量部分才能通过加法或减法合并。
| 3x and -7x | Like terms | 同类项 |
| 5y and 2y² | Unlike terms | 异类项 |
| 4ab and -2ab | Like terms | 同类项 |
| 6 and 9 | Like terms | 同类项 |
3. Coefficients and Variable Parts | 系数与变量部分
Every term can be split into a coefficient and a variable part. The coefficient is the numerical factor, while the variable part is the letter or letters, including any powers. In the term -5xy², the coefficient is -5 and the variable part is xy².
每一项都可以拆分为系数和变量部分。系数是数字因数，变量部分是一个或多个字母，包括任何指数。在项 -5xy² 中，系数是 -5，变量部分是 xy²。
When collecting like terms, keep the variable part unchanged. Only the coefficients are added or subtracted. This is because the variable part acts like a label or unit.
合并同类项时，变量部分保持不变。只需加减系数。这是因为变量部分就像标签或单位一样。
4. Collecting Like Terms Step by Step | 合并同类项分步法
To simplify an expression, first identify groups of like terms. It can help to underline or circle terms that have the same variable part. Then add or subtract the coefficients in each group.
要化简一个表达式，首先要找出同类项的组。可以在相同变量部分的项下面划线或画圈。然后将每组中的系数相加或相减。
nts are 2, 5, and -3. Adding them gives 2 + 5 - 3 = 4, so the simplified expression is 4x.
例如，化简 2x + 5x - 3x。这三项的变量部分都是 x。系数分别是 2、5 和 -3。相加得 2 + 5 - 3 = 4，所以化简结果为 4x。
2x + 5x - 3x = (2 + 5 - 3)x = 4x
5. Working with Constants | 处理常数项
Constants are terms with no variable part. For example, 6, -2, and 9 are constants. All constants are like terms with each other because their variable part is empty or can be seen as x⁰.
常数项是没有变量部分的项。例如 6、-2 和 9 都是常数。所有常数项彼此都是同类项，因为它们的变量部分为空，或者可以看作 x⁰。
s and the constants separately: 3x + 2x = 5x and 5 - 1 = 4. The simplified form is 5x + 4.
在表达式 3x + 5 + 2x - 1 中，分别合并 x 项和常数项：3x + 2x = 5x，5 - 1 = 4。化简结果是 5x + 4。
3x + 5 + 2x - 1 = 5x + 4
6. Powers Matter: x vs x² | 指数很重要：x 与 x²
A common IGCSE question asks students to simplify expressions such as x + x² + x. The terms x and x² may look similar, but they are not like terms. The powers are different, so they cannot be combined.
IGCSE 常见题目会要求学生化简 x + x² + x 这样的表达式。项 x 和 x² 看起来相似，但它们不是同类项。指数不同，因此不能合并。
The correct simplification is 2x + x². Write the answer in any order, but it is standard to put higher powers first: x² + 2x. The same idea applies to y and y³, or t² and t⁴.
正确的化简结果是 2x + x²。答案可以按任意顺序书写，但标准写法一般把较高次幂放在前面：x² + 2x。同样的规则适用于 y 和 y³，或 t² 和 t⁴。
e the power. The term x + x equals 2x, not x². Only multiplication such as x × x gives x².
还要记住，加法不会改变指数。x + x 等于 2x，而不是 x²。只有像 x × x 这样的乘法才会得到 x²。
7. Expressions with More Than One Variable | 含有多个变量的表达式
When an expression contains different variables, group each variable separately. For instance, simplify 2a + 3b + 5a - b. The a terms are 2a and 5a, giving 7a. The b terms are 3b and -b, giving 2b.
当表达式含有不同变量时，要分别合并每种变量。例如，化简 2a + 3b + 5a - b。a 项是 2a 和 5a，合并得 7a。b 项是 3b 和 -b，合并得 2b。
The simplified expression is 7a + 2b. You cannot combine 7a and 2b because the variable parts a and b are different. They are unlike terms.
化简结果是 7a + 2b。你不能把 7a 和 2b 合并，因为变量部分 a 和 b 不同。它们不是同类项。
Similarly, terms such as 3xy and 5x are not like terms. One has the variable part xy, while the other has only x. Always compare the full variable part before combining.
同样地，像 3xy 和 5x 这样的项不是同类项。一个的变量部分是 xy，而另一个只有 x。在合并之前，一定要比较完整的变量部分。
8. Removing Brackets Before Collecting | 先去括号再合并
the bracket: 3x + 6 + 2x. Then collect like terms: 3x + 2x = 5x, so the answer is 5x + 6.
有些表达式含有括号，必须先展开括号再合并同类项。例如 3(x + 2) + 2x。先将括号乘以 3：3x + 6 + 2x。然后合并同类项：3x + 2x = 5x，所以答案是 5x + 6。
If there is a negative sign in front of a bracket, multiply every term inside by -1. For example, 4x - (x + 3) becomes 4x - x - 3, which simplifies to 3x - 3.
如果括号前面是负号，要把括号内的每一项都乘以 -1。例如，4x - (x + 3) 变为 4x - x - 3，化简得 3x - 3。
For a bracket multiplied by a number, distribute that number to every term inside. In 2
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