Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, expressions are built from terms. Understanding how to combine like terms is essential for simplifying expressions, solving equations, and working with formulae. In this lesson, we will learn what like terms are, how to identify them, and how to combine them correctly.

在代数中，表达式由项构成。理解如何合并同类项对于简化表达式、解方程和使用公式至关重要。在本课中，我们将学习什么是同类项、如何识别它们以及如何正确地合并它们。

1. What Are Like Terms? | 什么是同类项？

A term is a product of a number (coefficient) and one or more variables raised to powers. For example, in the term 5x², 5 is the coefficient and x² is the variable part. Like terms are terms that have exactly the same variable part, including the same powers. Only the coefficients may differ.

项是一个数字（系数）与一个或多个变量的幂的乘积。例如，在项 5x² 中，5 是系数，x² 是变量部分。同类项是指变量部分完全相同的项，包括相同的幂。只有系数可以不同。

3x and 5x are like terms because both have the variable x.

3x 与 5x 是同类项，因为都含有变量 x。
4y² and -2y² are like terms because both have y².

4y² 与 -2y² 是同类项，因为都含有 y²。
3x and 3y are NOT like terms because the variables are different.

3x 与 3y 不是同类项，因为变量不同。
7 and 9 are like terms because both are constants (no variable).

7 与 9 是同类项，因为都是常数项（不含变量）。

2. Why Do We Combine Like Terms? | 为什么需要合并同类项？

Combining like terms makes an expression shorter and easier to work with. It is similar to grouping counters: 2 apples + 3 apples = 5 apples, but 2 apples + 3 oranges cannot be simplified. In algebra, combining like terms preserves the value of the expression and prepares it for further operations such as solving equations or evaluating the expression at a given value.

合并同类项使表达式更简洁，更易于运算。这类似于将计数器分组：2 个苹果 + 3 个苹果 = 5 个苹果，但 2 个苹果 + 3 个橙子不能简化。在代数中，合并同类项保持了表达式的值，并为进一步运算做好准备，例如解方程或在给定值处求表达式的值。

3. The Golden Rule of Like Terms | 合并同类项的黄金法则

Only terms with exactly the same variables and the same exponents can be combined. For example, x and x² are not like terms because the powers of x are different. A constant term (a number with no variable) is only like with other constants. The variable part, including all exponents, must match perfectly.

只有变量和指数完全相同的项才能合并。例如，x 和 x² 不是同类项，因为 x 的幂不同。常数项（没有变量的数字）只能与其他常数项合并。变量部分（包括所有指数）必须完全匹配。

Expression	Like terms?
6x + 2x	Yes – same variable x
5x + 5x²	No – different powers
4ab + 6ab	Yes – both have ab
3 + 7	Yes – both constants

4. Working with Coefficients | 系数的运算

To combine like terms, add or subtract the coefficients, and keep the variable part exactly the same. This is often called “collecting like terms”. The variable part acts as a common factor. For example, if you have 2x + 3x, the coefficient of x after combining is 2 + 3 = 5, so the result is 5x.

合并同类项时，需对系数进行加或减运算，并保持变量部分完全相同。这通常被称为“收集同类项”。变量部分作为公因数。例如，对于 2x + 3x，合并后 x 的系数为 2 + 3 = 5，因此结果是 5x。

2x + 3x = (2 + 3)x = 5x

Notice that we do not change the variable part. Only the number in front (the coefficient) changes.

注意我们不会改变变量部分。只有前面的数字（系数）发生变化。

5. Example: Combining Positive Terms | 例：正项合并

Let’s simplify 2x + 3x + 4x. All three terms are like terms because each has the same variable x. Add the coefficients: 2 + 3 + 4 = 9. Therefore, the simplified expression is 9x. This means that no matter what value x takes, the original expression and 9x are exactly equal.

让我们简化 2x + 3x + 4x。这三个项都是同类项，因为都含有相同的变量 x。将系数相加：2 + 3 + 4 = 9。因此，简化后的表达式为 9x。这意味着无论 x 取何值，原表达式与 9x 完全相等。

2x + 3x + 4x = (2 + 3 + 4)x = 9x

This process can be extended to any number of like terms. Just add the coefficients in order.

这个过程可以扩展到任意数量的同类项。只需按顺序将系数相加。

6. Example: Combining Negative and Positive Terms | 例：正负项合并

Simplify 5y − 2y + 8y − y. Note that −y is the same as −1y. Combine the coefficients in order: 5 − 2 + 8 − 1 = 10. So the result is 10y. Care with signs is crucial: subtracting a term is the same as adding its negative coefficient.

简化 5y − 2y + 8y − y。注意 −y 等同于 −1y。按顺序合并系数：5 − 2 + 8 − 1 = 10。因此结果是 10y。注意符号至关重要：减去一项等同于加上其相反数系数。

5y − 2y + 8y − y = (5 − 2 + 8 − 1)y = 10y

If you prefer, group positive and negative coefficients separately: (5 + 8) − (2 + 1) = 13 − 3 = 10, same answer.

你也可以将正负系数分别分组：(5 + 8) − (2 + 1) = 13 − 3 = 10，答案相同。

7. Combining Terms with Different Powers | 不同次幂的合并

Terms with different powers of the same variable cannot be combined. For example, x² and x are unlike terms. You must keep them separate. Simplify 2x² + 3x + 4x² − x. First group the x² terms: 2x² + 4x² = 6x². Then group the x terms: 3x − x = 2x. So the expression becomes 6x² + 2x.

同一变量的不同次幂的项不能合并。例如，x² 和 x 是不同类项。必须将它们分开。简化 2x² + 3x + 4x² − x。首先把 x² 项分组：2x² + 4x² = 6x²。然后把 x 项分组：3x − x = 2x。因此表达式变为 6x² + 2x。

2x² + 3x + 4x² − x = (2x² + 4x²) + (3x − x) = 6x² + 2x

Think of x² and x as different “families”. You would never add a square metre to a metre in measurement, and likewise you keep x² and x separate in algebra.

可以把 x² 和 x 视为不同的“家族”。在测量中你绝不会把平方米和米相加，同样在代数中也要把 x² 和 x 分开。

8. Combining Terms with Multiple Variables | 多项变量合并

When terms contain more than one variable, the entire variable part must match. For example, 3ab and 2ab are like terms, but 3ab and 2a are not. Simplify 3ab + 2a − 2ab + 4a: combine the ab terms (3 − 2)ab = ab; combine the a terms (2 + 4)a = 6a. The result is ab + 6a.

当项含有多个变量时，整个变量部分必须匹配。例如，3ab 和 2ab 是同类项，但 3ab 和 2a 不是。简化 3ab + 2a − 2ab + 4a：合并 ab 项 (3 − 2)ab = ab；合并 a 项 (2 + 4)a = 6a。结果为 ab + 6a。

3ab + 2a − 2ab + 4a = (3 − 2)ab + (2 + 4)a = ab + 6a

Notice that in ab, the individual variables a and b are linked by multiplication. To combine terms, the exact pair (or set) of variables must appear in every term.

注意在 ab 中，变量 a 和 b 通过乘法相连。要合并项，每个项中必须出现完全相同的变量组合。

9. Simplifying Before Combining | 合并前的化简

Sometimes you need to expand brackets first before collecting like terms. For example, simplify 2(x + 3) + 3(x − 1). First expand: 2x + 6 + 3x − 3. Now combine like x terms: 2x + 3x = 5x. Combine constants: 6 − 3 = 3. The simplified expression is 5x + 3.

有时在合并同类项之前需要先展开括号。例如，简化 2(x + 3) + 3(x − 1)。先展开：2x + 6 + 3x − 3。现在合并 x 项：2x + 3x = 5x。合并常数项：6 − 3 = 3。简化后的表达式为 5x + 3。

2(x + 3) + 3(x − 1) = 2x + 6 + 3x − 3 = 5x + 3

Always expand brackets completely before you start combining. Missing a term during expansion is a frequent source of errors.

在开始合并之前一定要完整地展开括号。展开时漏掉一项是常见错误来源。

10. Real-World Application | 实际应用

Combining like terms is useful in geometry and everyday problems. For example, the perimeter of a rectangle with length 3x + 2 and width 2x − 1 can be found by adding twice the length and twice the width: 2(3x + 2) + 2(2x − 1) = 6x + 4 + 4x − 2 = 10x + 2. This compact expression lets you calculate the perimeter for any value of x.

合并同类项在几何和日常问题中非常有用。例如，一个长为 3x + 2、宽为 2x − 1 的矩形，其周长可以通过两倍长加两倍宽来求得：2(3x + 2) + 2(2x − 1) = 6x + 4 + 4x − 2 = 10x + 2。这个简洁的表达式让你可以针对任意 x 值计算周长。

P = 2(3x + 2) + 2(2x − 1) = 6x + 4 + 4x − 2 = 10x + 2

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Combining Like Terms | 合并同类项