Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In IGCSE algebra, simplifying an expression often begins with collecting like terms. This skill turns messy sums such as 3x + 5y + 2x − y into the cleaner 5x + 4y, and it is essential before solving equations, expanding brackets or factorising.

在 IGCSE 代数中，化简表达式通常从合并同类项开始。这项技能可以把 3x + 5y + 2x − y 这样杂乱的式子整理成更简洁的 5x + 4y。在解方程、去括号或因式分解之前，合并同类项都是必不可少的基础。

1. What Are Like Terms? | 什么是同类项？

A term is a single number, letter, or product of numbers and letters. Like terms have exactly the same variable part, including the same powers. The number in front is called the coefficient, and coefficients may differ.

项是单个数字、字母或数与字母的乘积。同类项拥有完全相同的变量部分，包括相同的指数。项前面的数字称为系数，系数可以不同。

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.

例如，3x 和 5x 是同类项，因为它们都含有一次方的 x。然而，3x 和 3x² 不是同类项，因为 x 的指数不同。

3x + 5x = 8x

2. Identifying Like Terms | 识别同类项

To identify like terms, ignore the coefficient and focus on the letter part. Terms such as 2xy, −7xy and 0.5xy are like because the variable part xy is identical.

识别同类项时，先忽略系数，只看字母部分。像 2xy、−7xy 和 0.5xy 这样的项是同类项，因为变量部分 xy 完全相同。

Be careful: 4ab and 4a are not like terms; the first contains both a and b, while the second contains only a.

注意：4ab 和 4a 不是同类项；前者含有 a 和 b，后者只含有 a。

Quick test: In 7m + 3n − 2m + n, the like-term pairs are 7m and −2m, and 3n and n.

快速检验：在 7m + 3n − 2m + n 中，同类项配对是 7m 和 −2m，以及 3n 和 n。

3. The Basic Rule of Combining | 合并的基本规则

When combining like terms, add or subtract the coefficients and keep the variable part unchanged. This works because the variable part is a common factor.

合并同类项时，将系数相加或相减，变量部分保持不变。这是因为变量部分是公因数。

ax + bx = (a + b)x

Example: 3x + 2x = 5x, because 3 + 2 = 5. Similarly, 7y − 4y = 3y, because 7 − 4 = 3.

例如：3x + 2x = 5x，因为 3 + 2 = 5。同理，7y − 4y = 3y，因为 7 − 4 = 3。

4. Single-Variable Expressions | 单变量表达式

With one variable, collect all terms containing that variable, then collect the constant terms separately.

对于单变量表达式，先合并所有含该变量的项，再单独合并常数项。

Worked example: Simplify 5m + 3m − 2m. Add coefficients: 5 + 3 − 2 = 6, so the answer is 6m.

例题：化简 5m + 3m − 2m。系数相加：5 + 3 − 2 = 6，因此答案是 6m。

Another example: 8p − 3p + p. Remember p has coefficient 1. So 8 − 3 + 1 = 6, giving 6p.

另一个例子：8p − 3p + p。记住 p 的系数是 1。所以 8 − 3 + 1 = 6，得到 6p。

5. Multi-Variable Terms | 多变量项

For terms with more than one variable, every variable and every exponent must match. For instance, 3xy and 5xy are like, but 2ab and 3a are not.

对于含有多个变量的项，每个变量及其指数都必须完全匹配。例如，3xy 和 5xy 是同类项，但 2ab 和 3a 不是。

Worked example: Simplify 4xy + 2x + 7xy − x. Group 4xy + 7xy = 11xy and 2x − x = x, so the final expression is 11xy + x.

例题：化简 4xy + 2x + 7xy − x。把 4xy + 7xy 合并为 11xy，把 2x − x 合并为 x，所以最终表达式为 11xy + x。

Order of terms in the answer can be changed, but IGCSE answers usually keep alphabetical order or descending powers where possible.

答案中项的顺序可以调整，但 IGCSE 答案通常按字母顺序或次数从高到低排列。

6. Combining with Negative Coefficients | 含负系数的合并

Negative signs belong to the coefficient of the term. When moving terms around, keep the sign in front of each term.

负号属于该项

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Collecting Like Terms | 合并同类项