📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, simplifying algebraic expressions by collecting like terms is one of the first and most important algebra skills. It allows you to reduce long expressions into shorter, manageable forms and is essential for solving equations, expanding brackets, and working with formulas.
在IGCSE数学中，通过合并同类项化简代数表达式是最早也是最重要的代数技能之一。它能将较长的表达式简化为更短、更易处理的形式，对于解方程、去括号和运用公式都至关重要。
1. What Are Like Terms? | 什么是同类项
Like terms are terms that contain exactly the same variable letters, each raised to exactly the same powers. The numerical coefficients can be different, but the variable part must match.
同类项是指含有完全相同变量的项，且每个变量的指数也完全相同。数字系数可以不同，但变量部分必须一致。
For example, 3x and −5x are like terms because both contain the variable x raised to the power 1. Similarly, 2y² and 7y² are like terms because both contain y².
例如，3x 和 −5x 是同类项，因为二者都含有指数为1的变量 x。同样，2y² 和 7y² 是同类项，因为都含有 y²。
The term 4x²y and −x²y are also like terms: both contain x² and y. The coefficients 4 and −1 do not affect whether the terms are alike.
4x²y 和 −x²y 也是同类项：二者都含有 x² 和 y。系数 4 和 −1 不影响它们是否为同类项。
- 3x, −5x, x → all like terms
- 2y², 7y², −y² → all like terms
- 4x²y, −x²y → like terms
The list above shows groups of terms with the same variable part.
上面的列表显示了具有相同变量部分的项组。
2. Why Collecting Like Terms Matters | 为什么要合并同类项
Collecting like terms makes expressions shorter and easier to understand. In an exam, leaving an answer as 3x + 5x instead of 8x would often lose a simplification mark.
合并同类项可以让表达式更简短、更容易理解。在考试中，如果答案写成 3x + 5x 而不是 8x，通常会失去化简分。
It is also the foundation for later topics such as solving equations, rearranging formulas, and simplifying algebraic fractions. Without this skill, longer algebra problems become much harder.
它也是后续学习的基础，如解方程、变换公式和化简代数分式。没有这项技能，较长的代数问题会变得困难得多。
b)c. This is why the variable part stays the same and only the coefficients are combined.
合并同类项时，实际上是在逆向使用分配律：ac + bc = (a + b)c。这就是为什么变量部分保持不变，只合并系数。
3x + 5x = (3 + 5)x = 8x
In word problems, collecting like terms helps you write a simplified formula for total cost, perimeter, or distance.
在应用题中，合并同类项有助于写出总成本、周长或距离的简化公式。
3. Identifying Like Terms in Expressions | 在表达式中识别同类项
To identify like terms, ignore the sign and the coefficient first, and focus only on the variable part. The sign stays attached to the term, but it does not affect whether terms are like.
要识别同类项，先忽略符号和系数，只看变量部分。符号跟随该项，但不影响它们是否为同类项。
In the expression 4a + 3b − 2a + 7b, the like terms are 4a and −2a, and 3b and 7b.
在表达式 4a + 3b − 2a + 7b 中，同类项是 4a 和 −2a，以及 3b 和 7b。
| Term 1 | Term 2 | Like? |
| 7x | 2x | Yes |
| 7x | 2x² | No — powers differ |
| 5mn | 3mn | Yes |
| 5mn | 5m | No — one has n, one does not |
The table above helps you avoid grouping terms that look similar but have different variables or powers.
上表有助于避免将看似相似但变量或指数不同的项归为一组。
4. Single-Variable Collection | 单变量合并
When all terms contain the same variable, add or subtract the coefficients and keep the variable unchanged.
当所有项都含有相同变量时，将系数相加或相减，变量保持不变。
Simplify 6x + 9x − 2x:
化简
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