Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

Combining like terms is a fundamental skill in algebra that allows us to simplify expressions by adding or subtracting terms that have the same variable parts. Mastering this skill is essential for solving equations, factorising, and working with functions in IGCSE Mathematics.

合并同类项是代数中的一项基本技能，通过将具有相同变量部分的项相加或相减来简化表达式。掌握这一技能对于IGCSE数学中解方程、因式分解以及处理函数都至关重要。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable raised to the same power. Only the numerical coefficients may be different. For example, 3x and 5x are like terms because both contain the variable x to the power 1.

同类项是指具有完全相同变量且变量指数也相同的项。只有数字系数可能不同。例如，3x和5x是同类项，因为两者都含有一次幂的变量x。

Terms such as 4x² and 2x² are like terms, while 4x² and 2x are not like terms because the powers of x are different (2 versus 1).

像4x²和2x²这样的项是同类项，而4x²和2x不是同类项，因为x的幂不同（分别为2和1）。

Like terms: 2a, 5a, -3a

同类项：2a、5a、-3a
Like terms: 7xy² and 4xy²

同类项：7xy²和4xy²
Not like terms: 3x and 3x²

非同类项：3x和3x²
Not like terms: 2xy and 2x

非同类项：2xy和2x

2. Identifying Like Terms | 识别同类项

To identify like terms, ignore the coefficient and focus only on the variable part. The variable part must match exactly, including all powers and letters. The order of multiplication does not matter because multiplication is commutative: ab is the same as ba.

要识别同类项，忽略系数，只关注变量部分。变量部分必须完全一致，包括所有字母及其指数。乘法的顺序不影响结果，因为乘法满足交换律：ab与ba相同。

For example, 3ab and -4ba are like terms because ab equals ba. However, 3ab and 3a²b are not like terms because the power of a is different.

例如，3ab和-4ba是同类项，因为ab等于ba。但是，3ab和3a²b不是同类项，因为a的指数不同。

Expression \| 表达式	Like terms \| 同类项
5x + 3y – 2x + 7y	5x and -2x; 3y and 7y
4p² + 2p + p² – p	4p² and p²; 2p and -p
2abc + 3ab – 5abc	2abc and -5abc

3. The Distributive Law Foundation | 分配律基础

Combining like terms is based on the distributive law: a(b + c) = ab + ac. When we say 3x + 5x = 8x, we are applying the distributive law in reverse: x(3 + 5) = 8x. This reverse process is called factoring out the common variable.

合并同类项基于分配律：a(b + c) = ab + ac。当我们说3x + 5x = 8x时，我们是在逆向使用分配律：x(3 + 5) = 8x。这个逆向过程称为提取公因变量。

This is why we only add the coefficients: the variable part is common to both terms. The same logic applies to more complex variables like xy or m².

这就是为什么我们只相加系数：变量部分是两项所共有的。同样的逻辑适用于更复杂的变量，如xy或m²。

ax + bx = (a + b)x

ax² – bx² = (a – b)x²

4. Combining Terms with Addition and Subtraction | 合并加法和减法项

To simplify an expression containing several like terms, add or subtract their coefficients while keeping the variable part unchanged. Always treat the sign before each term as part of the term.

要简化包含多个同类项的表达式，只需将它们的系数相加或相减，保持变量部分不变。始终将每一项前面的符号视为该项的一部分。

Consider the expression 7m + 4 – 3m + 9. Group the like terms: m-terms: 7m – 3m = 4m. Constant terms: 4 + 9 = 13. The simplified expression is 4m + 13.

考虑表达式7m + 4 – 3m + 9。将同类项分组：m项：7m – 3m = 4m。常数项：4 + 9 = 13。简化后的表达式为4m + 13。

Let us work through another example: 6a + 2b – a – 5b. Grouping gives (6a – a) + (2b – 5b) = 5a – 3b.

再看另一个例子：6a + 2b – a – 5b。分组得到(6a – a) + (2b – 5b) = 5a – 3b。

5. Combining Terms with Powers | 合并含指数的项

Variable parts with indices must match exactly. x² and x³ cannot be combined because their powers differ. However, x² + 3x² = 4x² is valid because the exponent of x is identical (2).

带指数的变量部分必须完全匹配。x²和x³不能合并，因为它们的指数不同。然而，x² + 3x² = 4x²是有效的，因为x的指数相同（均为2）。

Be careful: when multiplying terms, indices add (x² × x³ = x⁵), but when adding or subtracting like terms, the index stays the same (x² + x³ cannot be simplified). These are two different operations that students often confuse.

注意：乘法运算中指数相加（x² × x³ = x⁵），但在加减同类项时指数保持不变（x² + x³不能化简）。这两种不同的运算常常被学生混淆。

Example: simplify 5y³ – 2y² + y³ + 4y². Grouping like terms: (5y³ + y³) + (-2y² + 4y²) = 6y³ + 2y².

示例：化简5y³ – 2y² + y³ + 4y²。合并同类项：(5y³ + y³) + (-2y² + 4y²) = 6y³ + 2y²。

6. Combining Terms with Products of Variables | 合并变量乘积项

Terms such as xy, 3xy, and -2xy are like terms because the variable part xy is identical. Terms such as xy and x²y are not like terms because the powers of x differ. Terms such as xy and xyz are not like terms because z appears only in the second.

像xy、3xy和-2xy这样的项是同类项，因为变量部分xy完全相同。像xy和x²y这样的项不是同类项，因为x的指数不同。像xy和xyz这样的项不是同类项，因为z只出现在第二项中。

Example: simplify 4pq – 3p + 2pq + 5p. Grouping: (4pq + 2pq) + (-3p + 5p) = 6pq + 2p.

示例：化简4pq – 3p + 2pq + 5p。分组：(4pq + 2pq) + (-3p + 5p) = 6pq + 2p。

Notice that 6pq and 2p cannot be combined further because their variable parts are different. The final answer is 6pq + 2p.

注意6pq和2p不能再合并，因为它们的变量部分不同。最终答案为6pq + 2p。

7. Expanding Brackets and Then Combining | 先去括号再合并

In many IGCSE questions, you must first expand brackets using the distributive law, then combine like terms. The order is always: expand first, then simplify.

在许多IGCSE题目中，你需要先用分配律去括号，然后再合并同类项。顺序始终是：先展开，再化简。

Example: simplify 3(x + 2) + 2(x – 1). First expand: 3x + 6 + 2x – 2. Then combine like terms: (3x + 2x) + (6 – 2) = 5x + 4.

示例：化简3(x + 2) + 2(x – 1)。先展开：3x + 6 + 2

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Combining Like Terms | 合并同类项