📚 Collecting Like Terms | 合并同类项
Collecting like terms is one of the most important simplification skills in algebra. When terms have exactly the same variable part, we can add or subtract their coefficients and write a simpler expression. This process is called collecting like terms.
合并同类项是代数化简中最基本、最重要的技巧之一。当几个项含有完全相同的字母部分时，我们可以把它们的系数相加或相减，从而得到一个更简洁的代数式。这个过程就叫做“合并同类项”。
1. The Language of Algebra | 代数的语言
An algebraic expression is made up of terms separated by plus or minus signs. For example, in the expression 5x + 3y − 2, the terms are 5x, 3y and −2. Each term carries the sign that is immediately in front of it.
代数式由若干“项”组成，项与项之间用加号或减号连接。例如，在式子 5x + 3y − 2 中，项分别是 5x、3y 和 −2。每一项都带着它前面的符号。
A coefficient is the number that multiplies a variable. In the term 4x, the coefficient is 4. In the term −6p², the coefficient is −6.
系数是变量前面的数字倍数。在项 4x 中，系数是 4；在项 −6p² 中，系数是 −6。
2. What Exactly Is a Term? | 什么是一个项？
A term can be a constant, a variable, or a product of numbers and variables. For instance, 7, x, 3x, −2xy and 5a²b are all terms.
一个项可以是常数、变量，也可以是数字与变量的乘积。例如 7、x、3x、−2xy 和 5a²b 都是项。
In the expression 2x + 4y − 6, the terms are 2x, 4y and −6. Notice that the constant −6 is also a term.
在式子 2x + 4y − 6 中，项为 2x、4y 和 −6。注意，常数 −6 也是一个项。
3. What Makes Terms “Like”? | 什么使项成为“同类”？
Terms are called like terms when their variable parts are exactly the same. The coefficient does not have to be the same, and the order of multiplication does not matter.
当两个项所含的字母部分完全相同时，它们就是“同类项”。同类项的系数不一定相同，字母相乘的先后顺序也不影响。
Like terms: 3x and −5x are like terms because both contain x.
同类项：3x 和 −5x 是同类项，因为它们都含有字母 x。
Like terms: 4xy and 7yx are like terms because xy = yx.
同类项：4xy 和 7yx 是同类项，因为 xy = yx（乘法交换律）。
Unlike terms: 4x and 4x² are not like terms because their powers of x are different.
非同类项：4x 与 4x² 不是同类项，因为 x 的幂不同。
4. The Hidden Coefficient 1 | 隐藏的系数 1
When a variable appears without a number in front of it, its coefficient is actually 1. So x means 1x, and −x means −1x.
当变量前面没有写数字时，它的系数实际上是 1。所以 x 就是 1x，−x 就是 −1x。
This hidden 1 is very important when collecting like terms:
这个隐藏的 1 在合并同类项时非常重要：
3x + x = 4x
Here, x is really 1x, so 3x + 1x = 4x.
这里 x 实际上是 1x，所以 3x + 1x = 4x。
5x − x = 4x
Again, 5x − 1x = 4x. Never forget the invisible coefficient 1.
同样，5x − 1x = 4x。千万不要忘记这个看不见的系数 1。
5. Combining Positive and Negative Like Terms | 合并正负同类项
To collect like terms, add or subtract only the coefficients, and keep the variable part unchanged. The sign of each term belongs to its coefficient.
合并同类项时，只对系数进行加减，字母部分保持不变。每一项自己的正负号都属于它的系数。
| Expression | Simplified | Reason |
| 7x + 2x | 9x | 7 + 2 = 9 |
| 5p − 8p | −3p | 5 − 8 = −3 |
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