Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, “combining like terms” is one of the most fundamental skills. It means simplifying an expression by adding or subtracting the coefficients of terms that have the same variable parts. Without this skill, solving equations, factorising, and working with algebraic fractions become unnecessarily difficult.

在代数中，“合并同类项”是最基础的技能之一。它是指通过将具有相同变量部分的项的系数相加或相减来简化表达式。没有这项技能，解方程、因式分解以及处理代数分式都会变得异常困难。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that contain exactly the same variable(s) raised to exactly the same power(s). For example, 3x and 5x are like terms because both contain x to the power 1. The coefficient (the number in front) can be different, but the variable part must match.

同类项是指包含完全相同变量且变量指数也完全相同的项。例如，3x 和 5x 是同类项，因为它们都含有 x 的一次幂。系数（前面的数字）可以不同，但变量部分必须一致。

On the other hand, 3x and 3x² are not like terms because the powers of x differ. Similarly, 2xy and 2x are not like terms because the variable parts are not identical (one has y, the other does not).

另一方面，3x 和 3x² 不是同类项，因为 x 的指数不同。同样，2xy 和 2x 也不是同类项，因为它们的变量部分不完全相同（一个有 y，另一个没有）。

2. Recognising Like Terms: Variables and Exponents | 识别同类项：变量与指数

To decide whether two terms are like terms, look at two things: the variables and their exponents. The variables must be the same letters, and each letter must have the same exponent in both terms. The order in which variables are written does not matter, for example 2xy and 3yx are like terms because multiplication is commutative.

要判断两个项是否为同类项，需要看两点：变量及其指数。变量必须是相同的字母，且每个字母在两个项中的指数必须相同。变量的书写顺序无关紧要，例如 2xy 和 3yx 是同类项，因为乘法满足交换律。

4a and -2a → like terms (both contain a¹)

4a 和 -2a → 同类项（都含 a¹）
7b² and 7b → not like terms (different exponents)

7b² 和 7b → 不是同类项（指数不同）
5ab and 5a → not like terms (variable parts differ)

5ab 和 5a → 不是同类项（变量部分不同）
6p²q and -2qp² → like terms (same letters, same exponents, order irrelevant)

6p²q 和 -2qp² → 同类项（字母相同，指数相同，顺序无关）

3. The Rule for Combining Like Terms | 合并同类项的规则

The rule is simple: add or subtract the coefficients, keep the variable part exactly unchanged. For example, 3x + 5x = (3 + 5)x = 8x. The variable x remains x; only the number in front changes.

规则很简单：对系数进行加或减，变量部分保持不变。例如，3x + 5x = (3 + 5)x = 8x。变量 x 仍然是 x，只有前面的数字改变。

For subtraction, the same rule applies: 9y − 4y = (9 − 4)y = 5y. If a term has no written coefficient, the coefficient is understood to be 1, so x + 2x = 1x + 2x = 3x.

对于减法，同样的规则适用：9y − 4y = (9 − 4)y = 5y。如果一个项没有书写系数，那么它的系数默认为 1，所以 x + 2x = 1x + 2x = 3x。

Simplify: 4a + 3a − 2a = (4 + 3 − 2)a = 5a

This process is exactly “combining like terms”. You never multiply or divide the variable part during combination.

这个过程就是“合并同类项”。在合并时，你绝不乘或除变量部分。

4. Worked Examples: Simple Monomials | 实例：简单单项式

Let us apply the rule to a few expressions with single-variable monomials.

让我们将规则应用于一些只含单一变量的单项式表达式。

Example A: Simplify 2x + 7x − 5x.

例 A：化简 2x + 7x − 5x。

All three terms are like terms. Combine coefficients: 2 + 7 − 5 = 4. So the simplified result is 4x.

这三个项都是同类项。合并系数：2 + 7 − 5 = 4。所以化简结果为 4x。

Example B: Simplify 3a² + 6a² − a².

例 B：化简 3a² + 6a² − a²。

Coefficient of the first term is 3, second is 6, third is −1. Sum: 3 + 6 − 1 = 8. Thus the answer is 8a².

第一项的系数是 3，第二项是 6，第三项是 −1。和：3 + 6 − 1 = 8。因此答案是 8a²。

5. Combining Like Terms in Polynomials with Brackets | 含括号的多项式合并

Often expressions contain brackets. The first step is to expand the brackets, then combine like terms. For example:

许多表达式含有括号。第一步是去括号，然后合并同类项。例如：

Simplify 3(2x + 1) + 2(4x − 5).

化简 3(2x + 1) + 2(4x − 5)。

First expand: 6x + 3 + 8x − 10. Then identify like terms: 6x and 8x are like; 3 and −10 are constants (also like terms because no variable part). Combine x-terms: 6x + 8x = 14x. Combine constants: 3 − 10 = −7. Final answer: 14x − 7.

先去括号：6x + 3 + 8x − 10。然后识别同类项：6x 和 8x 是同类；3 和 −10 是常数（因为它们没有变量部分，所以也是同类项）。合并 x 项：6x + 8x = 14x。合并常数：3 − 10 = −7。最终答案：14x − 7。

Brackets with a negative sign require careful sign changes. For instance, 2(3x − 4) − (x − 1) = 6x − 8 − x + 1. Now combine: 6x − x = 5x and −8 + 1 = −7, giving 5x − 7.

带有负号的括号需要小心地改变符号。例如，2(3x − 4) − (x − 1) = 6x − 8 − x + 1。现在合并：6x − x = 5x，−8 + 1 = −7，得到 5x − 7。

6. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Many students make mistakes when the exponents or multiple variables are involved. Let us examine typical errors.

许多学生在涉及指数或多个变量时会犯错。让我们来看一些典型错误。

Mistake: 2x + 3x² = 5x³. This is wrong because 2x and 3x² are not like terms; they cannot be combined.

错误：2x + 3x² = 5x³。这是错误的，因为 2x 和 3x² 不是同类项，不能合并。
Mistake: 4x + 2y = 6xy. Wrong: x and y are different variables, so the terms are not like. The expression remains 4x + 2y.

错误：4x + 2y = 6xy。错误：x 和 y 是不同的变量，所以这些项不是同类项。表达式仍为 4x + 2y。
Mistake: 5z − (2z − 3) = 5z − 2z − 3 = 3z − 3. Wrong: subtracting a bracket changes the sign of every term inside. Correct is 5z − 2z + 3 = 3z + 3.

错误：5z − (2z − 3) = 5z − 2z − 3 = 3z − 3。错误：减去括号会改变括号内每一项的符号。正确是 5z − 2z + 3 = 3z + 3。
Mistake: Treating squared terms as if they were linear. 4x² + x² is 5x², never 5x.

错误：把平方项当作一次项。4x² + x² 是 5x²，绝不是 5x。

To avoid these mistakes, always mark or circle the variable part of each term before combining. If the variable parts are not identical, do not combine them.

为避免这些错误，在合并前，请始终标记或圈出每项的变量部分。如果变量部分不完全相同，就不要合并。

7. Advanced Applications: Multiple Variables and Higher Powers | 进阶应用：多变量与高次幂

When expressions contain more than one variable, combine terms with identical variable–exponent pairs. For example:

当表达式包含多个变量时，请合并具有相同变量-指数对的项。例如：

Simplify: 3xy + 5x² − 2xy + 4y² − x²

Group the like terms: terms with xy: 3xy − 2xy = xy. Terms with x²: 5x² − x² = 4x². Terms with y²: 4y². The final expression is xy + 4x² + 4y².

将同类项分组：含 xy 的项：3xy − 2xy = xy。含 x² 的项：5x² − x² = 4x²。含 y² 的项：4y²。最终表达式为 xy + 4x² + 4y²。

Notice that we cannot combine xy with x² or y² because the variable combinations are different. Also, a term like 2x²y is not like 3xy², because the exponents on x and y are different (x²y has x², while xy² has y²).

注意，我们不能将 xy 与 x² 或 y² 合并，因为变量组合不同。同样，像 2x²y 这样的项与 3xy² 不是同类项，因为 x 和 y 的指数不同（x²y 是 x²，而 xy² 是 y²）。

For higher powers, treat x³ as a separate category from x and x². Always combine only matching powers.

对于更高次幂，把 x³ 视为与 x 和 x² 不同的类别。始终只合并指数匹配的项。

8. Practice Problems | 练习题

Try these exercises to test your understanding of combining like terms.

请尝试以下练习来检验你对合并同类项的理解。

1. 7m + 2m − 3m	2. 5p² + 3p² − 4p
3. 2(3x + 4) + (2x − 1)	4. 6ab − 2a + 3ab + 5a

Solutions: 1. 6m. 2. 8p² − 4p (note 4p cannot be combined with 8p²). 3. 6x + 8 + 2x − 1 = 8x + 7. 4. 6ab + 3ab = 9ab; −2a + 5a = 3a; so the answer is 9ab + 3a.

答案：1. 6m。2. 8p² − 4p（注意 4p 不能与 8p² 合并）。3. 6x + 8 + 2x − 1 = 8x + 7。4. 6ab + 3ab = 9ab；−2a + 5a = 3a；所以答案为 9ab + 3a。

Remember that constants (numbers without variables) are also like terms with each other. For example, 3 + 5 − 2 = 6. They should be combined separately from variable terms.

请记住，常数（没有变量的数字）之间也是同类项。例如，3 + 5 − 2 = 6。它们应该与变量项分别合并。

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Combining Like Terms | 合并同类项