📚 Collecting Like Terms: Simplify Algebraic Expressions | 合并同类项：化简代数式
In IGCSE algebra, simplifying expressions often starts with collecting like terms. A term is a single number, variable, or product such as 7, x, 3x, −2x², or 5xy. Collecting like terms means grouping terms that have exactly the same variable part and exponent, then adding or subtracting their coefficients. This skill is a foundation for solving equations, factorising, and rearranging formulae.
在 IGCSE 代数中，化简表达式通常从合并同类项开始。项是指单个数字、变量或乘积，例如 7、x、3x、−2x² 或 5xy。合并同类项就是把变量部分和指数完全相同的项归为一组，然后将它们的系数相加或相减。这项技能是解方程、因式分解和公式变形的基础。
1. What Are Like Terms? | 什么是同类项？
A term is a part of an algebraic expression separated by + or − signs. For example, in 4x + 3y − 7, the terms are 4x, 3y, and −7. Like terms have exactly the same variables raised to exactly the same powers; only the numerical coefficients may differ. Constants such as 5 and −3 are also like terms because they have no variable part.
项是由加号或减号分隔的代数式的一部分。例如，在 4x + 3y − 7 中，项是 4x、3y 和 −7。同类项具有完全相同的变量和完全相同的指数；只有数字系数可能不同。常数项如 5 和 −3 也是同类项，因为它们没有变量部分。
Examples of like terms are 3x and −5x, 2a² and 7a², and 4xy and −xy. Non-examples are 3x and 3x² because the powers of x differ, and 2xy and 3x because the variable parts differ.
同类项的例子有 3x 和 −5x、2a² 和 7a²、4xy 和 −xy。非同类项的例子有 3x 和 3x²，因为 x 的指数不同；以及 2xy 和 3x，因为变量部分不同。
2. Why Collect Like Terms? | 为什么要合并同类项？
Collecting like terms makes an expression shorter and easier to use. If you have 2x + 3x, it can be written as 5x. This is allowed because x is the same variable in both terms. Simpler expressions help you solve equations, substitute values, and compare algebraic forms.
合并同类项可以让表达式更短、更易使用。如果你有 2x + 3x，它可以写成 5x。这是可行的，因为两个项中的变量 x 相同。更简单的表达式有助于解方程、代入数值以及比较代数形式。
The value of the expression does not change when you collect like terms; only its appearance changes. For example, 4a + 3a and 7a represent the same quantity for every value of a.
合并同类项不会改变表达式的值，只会改变它的形式。例如，4a + 3a 和 7a 对于 a 的每一个取值都表示相同的量。
3. Identifying Like Terms in Simple Expressions | 在简单表达式中识别同类项
To identify like terms, ignore the coefficient and focus on the variable part. Write each variable in alphabetical order and check the exponent. Terms such as xy and yx are like terms because multiplication is commutative, so xy and yx refer to the same variable part.
要识别同类项，可以忽略系数，只关注变量部分。把每个变量按字母顺序书写，并检查指数。xy 和 yx 这样的项是同类项，因为乘法满足交换律，所以 xy 和 yx 表示相同的变量部分。
| Expression | 表达式 | Like-term groups | 同类项组 | Simplified | 化简结果 |
|---|---|---|
| 3a + 2b + 5a − b | 3a, 5a and 2b, −b | 8a + b |
| x² + 3x + 2x² − x | x², 2x² and 3x, −x | 3x² + 2x |
| 5 + 2m − 3 + 7m | 5, −3 and 2m, 7m | 9m + 2 |
4. Adding and Subtracting Like Terms | 合并同类项：加法与减法
5x. For terms with multiple variables, the variable part stays the same: 4ab + 3ab = 7ab.
合并同类项时，将系数相加或相减，变量部分保持不变。例如，7x − 2x = (7 − 2)x = 5x。对于含有多个变量的项，变量部分保持不变：4ab + 3ab = 7ab。
3x + 2y + 5x − y = (3x + 5x) + (2y − y) = 8x + y
When the expression contains more than one type of like term, group each type separately. In the example above, all x terms are grouped together and all y terms are grouped together.
当表达式中含有不止一种同类项时，应分别对每一类进行分组。在上面的例子中，所有含 x 的项被分为一组，所有含 y 的项被分为另一组。
5. Dealing with Different Variables and Powers | 处理不同变量与幂次
Only terms with identical variable parts can be combined. The terms x, x², and x³ are different, so 3x + 2x² cannot be simplified further. Similarly, xy and x are unlike terms because xy has an extra factor y. If an expression mixes terms such as 2x², 3x, and 5, treat each group separately.
只有变量部分完全相同的项才能合并。x、x² 和 x³ 是不同的项，因此 3x + 2x² 不能再化简。同样，xy 和 x 也不是同类项，因为 xy 多了一个因子 y。如果表达式中混合了 2x²、3x 和 5 等项，应分别处理每一组。
x², the x terms give 3x + x = 4x, and the constant is 7. The final answer is 2x² + 4x + 7.
例如，化简 4x² + 3x + 7 − 2x² + x。x
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图