📚 Combining Like Terms | IGCSE 数学：合并同类项
In IGCSE algebra, simplifying expressions often starts with collecting like terms. This topic is the foundation for solving equations, rearranging formulas, and working with graphs. Once you master the rules for combining terms, you can simplify expressions quickly and avoid common algebraic mistakes.
在 IGCSE 代数中，化简表达式通常从合并同类项开始。这个主题是解方程、变换公式和绘制图像的基础。一旦掌握了合并规则，你就能快速化简表达式，并避免常见的代数错误。
1. What Are Like Terms? | 什么是同类项？
In algebra, an expression is built from terms. A term can be a number, a variable, or a product such as 5x or -3xy. Like terms are terms whose variable part is exactly the same, including the same power. For example, 4x and -7x are like terms, while 4x and 4x² are not.
在代数中，表达式由项组成。项可以是数字、变量或乘积，例如 5x 或 -3xy。同类项是指变量部分完全相同的项，包括指数也相同。例如，4x 和 -7x 是同类项，而 4x 和 4x² 不是同类项。
Constants are also like terms with each other. For instance, 8 and -3 can be combined because neither has a variable part.
常数项之间也属于同类项。例如，8 和 -3 可以合并，因为它们都没有变量部分。
2. The Golden Rule: Same Variable and Same Power | 黄金法则：相同变量且相同指数
The golden rule for collecting like terms is simple: only combine terms that have exactly the same variable letters and exactly the same powers. You can think of the variable part as a label. If two terms have different labels, they cannot be combined directly.
合并同类项的黄金法则很简单：只合并变量字母和指数都完全相同的项。你可以把变量部分看作一个标签。如果两个项的标签不同，就不能直接合并。
Only like terms can be added or subtracted: 3x + 2x = 5x, but 3x + 2y cannot be simplified further.
只有同类项才能相加或相减：3x + 2x = 5x，但 3x + 2y 不能继续化简。
3. How to Identify Like Terms | 如何识别同类项
Use this table to decide quickly whether terms can be combined. Look carefully at both the variable letter and its power.
使用下面的表格可以快速判断项是否能合并。要仔细观察变量字母及其指数。
| First term | 第一项 | Second term | 第二项 | Like terms? | 是否同类？ | Reason | 原因 |
|---|---|---|---|
| 5x | 7x | Yes 是 | Same variable x, same power 1 |
| 5x | 7y | No 否 | Different variable |
| 4x² | -2x² | Yes 是 | Same x² |
| 4x² | 4x | No 否 | Powers differ |
| 3xy | -xy | Yes 是 | Same product xy |
| 3xy | 3x²y | No 否 | x has different power |
| 9 | -4 | Yes 是 | Both constants |
When identifying like terms, always include the sign and the full variable part. A quick visual check of the letters and powers is usually enough.
识别同类项时，一定要把符号和完整变量部分都考虑进去。通常快速检查字母和指数就足够了。
4. Basic Combining: One Variable | 基础合并：单一变量
When all terms have the same variable, add or subtract the coefficients only. The variable part stays unchanged. For example, 3x + 5x means three x’s plus five x’s, giving eight x’s.
当所有项都具有相同变量时，只需对系数进行加减，变量部分保持不变。例如，3x + 5x 表示三个 x 加五个 x，结果是八个 x。
ax + bx = (a + b)x
Example: 7y – 2y + y = (7 – 2 + 1)y = 6y
示例：7y – 2y + y = (7 – 2 + 1)y = 6y
Notice that y on its own has an invisible coefficient of 1. Always include this coefficient when combining.
注意，单独的 y 前面有一个隐形的系数 1。合并时一定要把这个系数算进去。
5. Combining Terms with Multiple Variables | 合并含多个变量的项
For terms with more than one variable, such as xy or ab, treat the entire variable product as one block. Combine only if the whole block matches exactly.
对于含有多个变量的项，例如 xy 或 ab，要把整个变量乘积当作一个整体。只有当整个乘积完全一致时才能合并。
Example: 3xy + 5xy = 8xy, because both terms have the same block xy. However, 3xy + 3x cannot be combined because the second term is missing y.
示例：3xy + 5xy = 8xy，因为两项都含有相同的整体 xy。但是，3xy + 3x 不能合并，因为第二项缺少 y。
Example: 2ab + 3a + 4ab = 6ab + 3a. The ab terms combine, but 3a remains separate.
示例：2ab + 3a + 4ab = 6ab + 3a。ab 项可以合并，但 3a 保持不动。
6. Dealing with Constants | 处理常数项
Constants are numbers without variables. Collect all positive and negative numbers at the end. Remember that the sign in front of a term belongs to that term.
常数项是没有变量的数字。将所有正数和负数常数项放在一起计算。记住：项前面的符号属于该项。
Example: 5x + 3 + 2x – 7. Group the x terms: 5x + 2x = 7x. Group the constants: 3 – 7 = -4. The simplified expression is 7x – 4.
示例：5x + 3 + 2x – 7。将 x 项分组：5x + 2x = 7x。将常数项分组：3 – 7 = -4。化简后的表达式为 7x – 4。
When rearranging terms, keep each sign attached. For example, -7 should be moved as -7, not as +7.
重新排列各项时，要让符号跟随每一项。例如，-7 移动时仍应写作 -7，而不是 +7。
7. Combining Terms with Powers | 合并含幂的项
Terms with powers are like only if the base and the exponent match exactly. So x and x² are not like terms, and x²y and xy² are not like terms. Combine coefficients without changing the exponent.
含有幂的项只有在底数和指数都完全相同时才是同类项。因此，x 和 x² 不是同类项，x²y 和 xy² 也不是同类项。合并系数时不要改变指数。
Example: 2x² + 5x² – x² = 6x². The expression 2x² + 3x cannot be simplified further because the powers differ.
示例：2x² + 5x² – x² = 6x²。表达式 2x² + 3x 不能继续化简，因为指数不同。
Example: 4x²y + 3xy² – x²y = 3x²y + 3xy². The x²y terms combine, but xy² is different because the powers are swapped.
示例：4x²y + 3xy² – x²y = 3x²y + 3xy²。x²y 项可以合并，但 xy² 因指数位置不同而不能合并。
8. Common Mistakes to Avoid | 常见错误与避免方法
Mistake 1: Combining unlike terms. For example, writing x + x² = 2x² is incorrect because x and x² have different powers.
错误 1：合并不相同的项。例如，把 x + x² 写成 2x² 是错误的，因为 x 和 x² 的指数不同。
Mistake 2: Losing the sign. In 5 – 2x, the minus sign belongs to 2x, so you cannot rewrite it as 5 + 2x.
错误 2：丢失符号。在 5 – 2x 中，减号属于 2x，所以不能把它改写成 5 + 2x。
Mistake 3: Changing the power when adding coefficients. The expression 3x² + 4x² is 7x², not 7x⁴.
错误 3：合并系数时改变了指数。3x² + 4x² 等于 7x²，而不是 7x⁴。
Mistake 4: Forgetting invisible coefficients. The term y has coefficient 1, so 3y + y = 4y.
错误 4：忘记隐形系数。y 的系数是 1，所以 3y +
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