Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, we often meet expressions that contain several terms. Some of these terms share exactly the same variable part, and we can add or subtract them together. This process is called collecting like terms, and it is one of the most basic skills in simplifying algebraic expressions.

在代数中，我们经常会遇到包含多个项的表达式。其中有些项具有完全相同的变量部分，我们可以将它们合并在一起。这一过程称为合并同类项，是化简代数表达式最基本的技能之一。

1. What are Terms? | 什么是项

An algebraic term is a number, a variable, or a product of a number and one or more variables. For example, in the expression 4x + 3y – 7, the terms are 4x, 3y and -7.

代数项是一个数、一个变量，或者一个数与一个或多个变量的乘积。例如，在表达式 4x + 3y – 7 中，各项是 4x、3y 和 -7。

Each term has two main parts: the coefficient (the number in front) and the variable part. In 4x, the coefficient is 4 and the variable part is x. In -7, there is no variable, so we call it a constant term.

每一项有两个主要部分：系数（前面的数字）和变量部分。在 4x 中，系数是 4，变量部分是 x。在 -7 中，没有变量，所以我们称它为常数项。

• The coefficient can be positive or negative. | 系数可以是正数或负数。

• A term such as x has an invisible coefficient of 1. | 像 x 这样的项其实隐藏着系数 1。

2. Understanding Like Terms | 理解同类项

Like terms are terms that have exactly the same variable part. This means the same letters raised to the same powers. For example, 5x and -2x are like terms because both have the variable x.

同类项是指变量部分完全相同的项。这意味着相同的字母且指数也相同。例如，5x 和 -2x 是同类项，因为它们都含有变量 x。

However, 3x and 3x² are not like terms, because the variable parts x and x² are different. The power of x is not the same.

但是，3x 和 3x² 不是同类项，因为变量部分 x 和 x² 不同，x 的指数不一样。

Notice that the order of multiplication does not matter: 3xy and -2yx are like terms, because xy = yx.

注意乘法的顺序无关紧要：3xy 和 -2yx 是同类项，因为 xy = yx。

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

To collect like terms, keep the variable part unchanged and add or subtract the coefficients only.

合并同类项时，保持变量部分不变，只对系数进行加减。

ax + bx = (a + b)x

For example, 3x + 5x = 8x. The variable x stays the same, and we add the coefficients 3 and 5.

例如，3x + 5x = 8x。变量 x 保持不变，我们把系数 3 和 5 相加。

If the terms are subtracted, use the same rule: 9y – 4y = 5y.

如果是相减，同样适用：9y – 4y = 5y。

4. Working with Addition and Subtraction | 加减运算

When an expression contains several like terms, group them first and then combine. For example, 2x + 3 – 5x + 8 can be rearranged as (2x – 5x) + (3 + 8) = -3x + 11.

当一个表达式中包含多个同类项时，先分组再合并。例如，2x + 3 – 5x + 8

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