📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, collecting like terms is the process of simplifying an expression by adding or subtracting terms that have exactly the same variable part. It is used constantly when solving equations, rearranging formulas and working with graphs, so it is one of the most important foundation skills in the course.
在 IGCSE 代数中，合并同类项是通过加减变量部分完全相同的项来化简表达式。解方程、变换公式和处理图像时都会经常用到它，因此它是本课程中最重要的基础技能之一。
1. What Are Like Terms? | 什么是同类项
Like terms are algebraic terms that contain exactly the same variables raised to exactly the same powers. The coefficients, or numbers in front of the variables, can be different. For example, 3x and −5x are like terms because both contain the variable x to the power 1.
同类项是指所含变量完全相同且变量指数也完全相同的代数项。系数（变量前面的数）可以不同。例如，3x 和 −5x 是同类项，因为两者都含有变量 x 的一次方。
Terms such as 2x and 3x² are not like terms, because the variable x is raised to different powers. Similarly, 4xy and 7x are not like terms, because the variable parts xy and x are different. The key question is always: do the terms have exactly the same letters with exactly the same powers?
像 2x 和 3x² 这样的项不是同类项，因为变量 x 的指数不同。同样，4xy 和 7x 也不是同类项，因为变量部分 xy 和 x 不同。关键问题始终是：这些项是否具有完全相同的字母和完全相同的指数？
2. Why Do We Collect Like Terms? | 为什么要合并同类项
An expression may contain several similar terms mixed together. Collecting like terms makes the expression shorter, clearer and easier to use in later calculations. For example, 2x + 5y + 3x − 2y can be simplified to 5x + 3y.
一个表达式可能包含多个混合在一起的相似项。合并同类项可以使表达式更短、更清晰，也便于后续计算。例如，2x + 5y + 3x − 2y 可以化简为 5x + 3y。
This simplification does not change the value of the expression. It only combines the amounts that are counted in the same way. In IGCSE questions, you are often asked to simplify expressions before substituting values. If the expression is not simplified, substitution can be longer and more likely to cause arithmetic errors.
这种化简不会改变表达式的值。它只是把用相同方式计数的量合并起来。在 IGCSE 题目中，通常会要求先化简表达式再代入数值。如果表达式没有化简，代入过程会更长，也更容易出现算术错误。
3. Recognising Like Terms with One Variable | 识别含一个变量的同类项
When an expression contains only one variable, you simply add or subtract the coefficients. For example, 4a + 7a − 2a means 4 groups of a plus 7 groups of a minus 2 groups of a.
当一个表达式只含有一个变量时，只需加减系数。例如，4a + 7a − 2a 表示 4 组 a 加上 7 组 a 再减去 2 组 a。
4a + 7a − 2a = (4 + 7 − 2)a = 9a
side gives 9 × 1 = 9. This is a useful way to test whether your simplification is correct.
你可以代入一个 a 的值来检验，例如 a = 1。左边得到 4 + 7 − 2 = 9，右边得到 9 × 1 = 9。这是检验化简是否正确的有效方法。
4. Constants and Terms Without Variables | 常数项与不含变量的项
A constant term is a number with no variable attached, such as 6, −3 or 1/2. Constants are like terms with each other, so all constant terms can be added or subtracted together.
常数项是不带变量的数，例如 6、−3 或 1/2。常数项彼此是同类项，因此所有常数项可以相加或相减。
For example, in the expression 5x + 3 + 2x − 8, the x terms are 5x and 2x, while the constant terms are 3 and −8. They simplify to 7x − 5.
例如，在表达式 5x + 3 + 2x − 8 中，含 x 的项是 5x 和 2x，常数项是 3 和 −8。它们化简为 7x − 5。
5x + 3 + 2x − 8 = (5x + 2x) + (3 − 8) = 7x − 5
Notice that the constant group is handled separately from the x group. Mixing them would be like adding apples and oranges, so the final answer keeps the variable part and the number part distinct.
请注意，常数项组与 x 项组是分开处理的。把它们混在一起就像把苹果和橙子相加，因此最终答案中变量部分和数字部分仍然保持独立。
5. Combining Powers of the Same Variable | 合并同一变量的幂
Only terms with the same power can be combined. For example, x and x² are different variable parts: x means x¹, while x² means x multiplied by itself. Therefore, 3x + 4x² cannot be simplified into a single term.
只有同次幂的项才能合并。例如，x 和 x² 是不同的变量部分：x 表示 x¹，而 x² 表示 x 自乘。因此，3x + 4x² 不能化简为单个项。
You can still simplify the powers separately. In 2x² + 3x − x² + 5x, the x² terms are 2x² and −x², and the x terms are 3x and 5x.
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