Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, simplifying expressions is a core skill. One of the most important techniques is combining like terms — the process of adding or subtracting terms that have the same variable parts. This article explains what like terms are, how to combine them, and why this skill matters in solving equations.

在代数中，化简表达式是一项核心技能。其中最重要的技巧之一就是合并同类项——即将具有相同变量部分的项进行相加或相减。本文将解释什么是同类项、如何合并它们，以及这一技能在解方程中的重要性。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variables raised to the same powers. The coefficients (the numbers in front) can be different. For example, \(3x\) and \(5x\) are like terms because both have the variable \(x\) to the first power.

同类项是指具有完全相同变量且变量指数相同的项。系数（项前面的数字）可以不同。例如，\(3x\) 和 \(5x\) 是同类项，因为它们都含有一次的变量 \(x\)。

Here are more examples of like terms:

以下是更多同类项的例子：

\(2a\) and \(7a\)
\(4x²\) and \(-3x²\)
\(-5xy\) and \(9xy\)
\(6\) and \(-8\) (constant terms are like terms)

\(2a\) 与 \(7a\)
\(4x²\) 与 \(-3x²\)
\(-5xy\) 与 \(9xy\)
\(6\) 与 \(-8\)（常数项也是同类项）

2. Identifying Like Terms | 识别同类项

To identify like terms, look at the variable part only. The variable part includes both the letters and their exponents. Terms are like only if the variable part is identical. For example, \(x\) and \(x²\) are not like terms because the exponents differ.

识别同类项时，只需关注变量部分。变量部分包括字母及其指数。只有当变量部分完全相同时，项才是同类项。例如，\(x\) 和 \(x²\) 不是同类项，因为指数不同。

Consider the expression: \(4x + 3y – 2x + 5\). Which terms can be combined?

看这个表达式：\(4x + 3y – 2x + 5\)。哪些项可以合并？

\(4x\) and \(-2x\) are like terms (both have \(x\)).
\(3y\) stands alone.
\(5\) is the only constant term.

\(4x\) 与 \(-2x\) 是同类项（都含有 \(x\)）。
\(3y\) 单独保留。
\(5\) 是唯一的常数项。

So the simplified expression is \(2x + 3y + 5\).

因此化简后的表达式为 \(2x + 3y + 5\)。

3. The Distributive Property | 乘法分配律

Before combining like terms, you may need to remove brackets using the distributive property. The rule is \(a(b+c) = ab + ac\). For example: \(3(2x + 4) = 3×2x + 3×4 = 6x + 12\).

在合并同类项之前，可能需要用乘法分配律去掉括号。规则是 \(a(b+c) = ab + ac\)。例如：\(3(2x + 4) = 3×2x + 3×4 = 6x + 12\)。

After expanding, you can then combine any like terms that appear.

展开后，就可以合并出现的同类项了。

\(a(b+c) = ab + ac\)

Be careful with negative signs: \(-2(3x – 5) = -6x + 10\).

注意负号：\(-2(3x – 5) = -6x + 10\)。

4. Combining Coefficients | 合并系数

When you combine like terms, you add or subtract their coefficients. The variable part stays exactly the same. For example, \(4x + 7x = (4+7)x = 11x\).

合并同类项时，将它们的系数相加或相减，变量部分完全保持不变。例如，\(4x + 7x = (4+7)x = 11x\)。

Similarly, \(9a² – 4a² = (9-4)a² = 5a²\).

类似地，\(9a² – 4a² = (9-4)a² = 5a²\)。

\(cx + dx = (c+d)x\)

This rule works for any variable or power, as long as the variable parts match.

只要变量部分相同，这个规则适用于任何变量或指数。

5. Combining Constants | 合并常数项

Constant terms have no variable part. They are always like terms with each other. For example, \(6 – 3 + 2 = 5\).

常数项没有变量部分。它们彼此之间总是同类项。例如，\(6 – 3 + 2 = 5\)。

In an expression like \(2x + 5 + 3x – 2\), combine the constants \(5 – 2 = 3\). Then combine the \(x\) terms: \(2x + 3x = 5x\). The result is \(5x + 3\).

在表达式 \(2x + 5 + 3x – 2\) 中，先合并常数项 \(5 – 2 = 3\)，再合并 \(x\) 项：\(2x + 3x = 5x\)。结果是 \(5x + 3\)。

Always keep the sign in front of each term when moving terms around.

在移动项时，要始终保留该项前面的符号。

6. Combining in Expressions | 表达式中的合并

An expression is a mathematical phrase without an equals sign. Simplifying expressions makes them shorter and easier to evaluate. For example, simplify \(3a + 2b – a + 4b\).

表达式是不含等号的数学短语。化简表达式使其更简短、更容易求值。例如，化简 \(3a + 2b – a + 4b\)。

Combine \(a\) terms: \(3a – a = 2a\).
Combine \(b\) terms: \(2b + 4b = 6b\).
Final expression: \(2a + 6b\).

合并 \(a\) 项：\(3a – a = 2a\)。
合并 \(b\) 项：\(2b + 4b = 6b\)。
最终表达式：\(2a + 6b\)。

Remember to write the terms in a consistent order, often alphabetical, but this is not required.

注意按一致的顺序书写各项，通常按字母顺序，但这并非强制要求。

7. Combining in Equations | 方程中的合并

When solving equations, combining like terms on each side simplifies the equation. For example: \(5x + 2x – 3 = 18\).

解方程时，先合并方程两边的同类项可以简化方程。例如：\(5x + 2x – 3 = 18\)。

First combine \(5x + 2x = 7x\). The equation becomes \(7x – 3 = 18\). Then add 3 to both sides: \(7x = 21\). Divide by 7: \(x = 3\).

先合并 \(5x + 2x = 7x\)，方程变为 \(7x – 3 = 18\)。然后两边加 3：\(7x = 21\)。两边除以 7：\(x = 3\)。

\(5x + 2x – 3 = 18 \rightarrow 7x – 3 = 18 \rightarrow x = 3\)

Combining first reduces the number of steps and avoids arithmetic errors.

先合并可以减少步骤并避免算术错误。

8. Combining with Multiple Variables | 多变量合并

Terms with different variables cannot be combined. For example, \(4x + 3y\) cannot be simplified further because \(x\) and \(y\) are different variables.

不同变量的项不能合并。例如，\(4x + 3y\) 无法进一步化简，因为 \(x\) 和 \(y\) 是不同的变量。

However, terms like \(2xy\) and \(5xy\) can be combined because the variable part \(xy\) is identical. Also, \(x²y\) and \(-3x²y\) are like terms.

但是，像 \(2xy\) 和 \(5xy\) 这样的项可以合并，因为变量部分 \(xy\) 相同。同样，\(x²y\) 和 \(-3x²y\) 也是同类项。

Variables	Like terms?
\(xy\) and \(x²y\)	No (different exponents)
\(x²y\) and \(-2x²y\)	Yes

注意变量的顺序不影响判断：\(xy\) 与 \(yx\) 是同类项，因为乘法交换律。

Note that order does not matter: \(xy\) and \(yx\) are like terms because of the commutative property of multiplication.

9. Common Mistakes | 常见错误

Below are typical mistakes students make when combining like terms, and how to avoid them.

以下是学生在合并同类项时经常犯的错误，以及如何避免它们。

Mistake	Example	Correct
Combining \(x\) and \(x²\)	\(3x + 2x² = 5x³\)	Cannot combine; leave as \(3x + 2x²\)
Forgetting the sign	\(5x – 3x = 8x\)	\(5x – 3x = 2x\)
Combining different variables	\(2x + 3y = 5xy\)	Leave as \(2x + 3y\)

Always check that the variable parts match exactly before combining.

合并前务必检查变量部分是否完全一致。

10. Practice Problems | 练习

Try these to build your confidence.

试试这些题目来建立信心。

Simplify \(7x + 2x – 5x\).
Simplify \(3a + 4b – a + 2b\).
Simplify \(2(x + 3) + 4x\).
Solve \(4x + 3x – 6 = 8\).

化简 \(7x + 2x – 5x\)。
化简 \(3a + 4b – a + 2b\)。
化简 \(2(x + 3) + 4x\)。
解方程 \(4x + 3x – 6 = 8\)。

Answers: 1. \(4x\)　2. \(2a + 6b\)　3. \(6x + 6\)　4. \(x = 2\)

答案：1. \(4x\)　2. \(2a + 6b\)　3. \(6x + 6\)　4. \(x = 2\)

11. Summary | 小结

Combining like terms is an essential algebra technique. Remember these key points:

合并同类项是一项重要的代数技巧。记住这些关键点：

Like terms have the same variable part and the same exponents.
Only add or subtract the coefficients; keep the variable part unchanged.
Constant terms are like terms with each other.
Use the distributive property to remove brackets before combining if needed.
Never combine different variables or different powers.

同类项具有相同的变量部分和相同的指数。
只对系数进行加减，变量部分保持不变。
常数项彼此之间都是同类项。
需要时先用分配律去掉括号，再合并。
绝不能合并不同变量或不同指数的项。

With consistent practice, combining like terms becomes quick and natural, helping you solve equations and simplify expressions with confidence.

通过持续练习，合并同类项会变得快速而自然，帮助你有信心地解方程和化简表达式。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Combining Like Terms | 合并同类项