Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

One of the first major skills you will master in algebra is collecting like terms. It is the process of simplifying algebraic expressions by combining terms that have the same variable part. Once you understand this, solving equations and manipulating formulae becomes much easier.

你将在代数中掌握的第一项重要技能就是合并同类项。它是指将变量部分相同的项合并起来，从而化简代数式。一旦理解了它，解方程和公式变形就会变得容易得多。

1. What Are Like Terms? | 什么是同类项？

In algebra, the term “like terms” refers to terms that have exactly the same variable part, including the same powers. The coefficient (the number in front of the variable) can be different. For example, 3x and -5x are like terms because both contain the variable x. 4x² and 7x² are like terms, but 4x² and 7x are not like terms because the powers of x differ.

在代数中，“同类项”指变量部分完全相同（包括指数相同）的项。变量前的系数可以不同。例如，3x 和 -5x 是同类项，因为都含有变量 x。4x² 和 7x² 是同类项，但 4x² 和 7x 不是同类项，因为 x 的指数不同。

Here are some examples to help you identify like terms:

下面是一些帮助你识别同类项的例子：

• Like terms: 2y and 5y; -3x² and x²; 8 and -4 (constants are like terms). | 同类项：2y 和 5y；-3x² 和 x²；8 和 -4（常数项也是同类项）。
• Unlike terms: 2y and 2y²; 3xy and 3x; 5 and 5x. | 不同类项：2y 和 2y²；3xy 和 3x；5 和 5x。

2. Why Do We Collect Like Terms? | 为什么要合并同类项？

Collecting like terms is essential because it simplifies expressions. A shorter expression is easier to evaluate, factor, and use in equations. It also helps you see the structure of a problem more clearly, and prevents errors in later calculations.

合并同类项至关重要，因为它能化简表达式。更简洁的表达式更容易求值、因式分解，也更便于在方程中使用。它还能帮助你更清楚地看出问题的结构，避免后续计算出错。

For example, the expression 5x + 3x – x is more complex than its simplified form 7x. By reducing the number of terms, you reduce the chance of making mistakes.

例如，表达式 5x + 3x – x 比其化简后的形式 7x 更复杂。通过减少项数，你可以降低出错的可能性。

When solving equations, collecting like terms is often the first step. For instance, in the equation 3x + 2x – 5 = 20, you can combine 3x and 2x into 5x, giving 5x – 5 = 20. This makes the next step of solving much clearer.

在解方程时，合并同类项通常是第一步。例如，在方程 3x + 2x – 5 = 20 中，你可以把 3x 和 2x 合并为 5x，得到 5x – 5 = 20。这样下一步求解就更加清晰。

3. How to Identify Like Terms | 如何识别同类项

To decide whether terms are like terms, ignore the coefficient and compare the variable part. The variable part contains the letters and their exponents. If two terms have exactly the same letters with the same exponents, they are like terms. For example, 2ab and -3ab are like terms, while 2ab and 2a²b are not.

要判断项是否为同类项，可以忽略系数，比较变量部分。变量部分包含字母及其指数。如果两项的字母和指数完全相同，它们就是同类项。例如，2ab 和 -3ab 是同类项，而 2ab 和 2a²b 不是。

The table below shows how to identify like and unlike terms:

下表展示了如何判断同类项和不同类项：