📚 Combining Like Terms | 合并同类项
Combining like terms is one of the most important skills in algebra. It is the process of simplifying an expression by adding or subtracting terms that have the same variable part. Without this skill, solving equations and working with graphs become much more difficult.
合并同类项是代数中最重要的技能之一。它通过把字母部分相同的项相加或相减来化简表达式。没有这项技能，解方程和函数图象的学习会变得困难得多。
1. What Are Like Terms? | 什么是同类项
An algebraic expression is made up of terms. In the expression 3x + 2y – 5, the terms are 3x, 2y and -5. The number in front of a variable is called the coefficient. Here, 3 and 2 are coefficients, while -5 is a constant term.
一个代数表达式由若干“项”组成。在表达式 3x + 2y – 5 中，各项是 3x、2y 和 -5。变量前面的数字称为系数。这里 3 和 2 是系数，而 -5 是常数项。
Two terms are like terms when they have exactly the same variables raised to the same powers. The coefficients do not have to be the same. For example, 4x and -7x are like terms because both have the variable x to the first power.
当两个项含有完全相同的字母，并且字母的指数也相同时，它们就是同类项。系数不一定相同。例如，4x 与 -7x 含有相同的变量 x，且都为一次幂，因此它们是同类项。
Terms that are not like terms cannot be combined. For example, 5x and 5y are not like terms because their variables are different.
不是同类项的项不能合并。例如，5x 与 5y 的变量不同，因此不是同类项。
| Expression | Terms | Can we combine? |
| 5x + 3x | 5x, 3x | Yes, both have x |
| 5x + 3x² | 5x, 3x² | No, powers differ |
| 5x + 3y | 5x, 3y | No, variables differ |
2. Why Do We Combine Like Terms? | 为什么合并同类项
Combining like terms makes expressions shorter and easier to understand. It also prepares an expression for substitution or equation solving.
合并同类项能让表达式更简短、更清晰，也为代入求值或解方程做好准备。
For example, 2x + 3 + 4x + 5 can be simplified by adding the x-terms and the constants separately:
例如，2x + 3 + 4x + 5 可以分别合并含 x 的项和常数项：
2x + 4x + 3 + 5 = 6x + 8
This simplified form is much easier to evaluate when a value of x is known.
化简后的形式在已知 x 的值时更容易计算。
3. The Distributive Property Connection | 与乘法分配律的联系
Combining like terms is actually an application of the distributive property. Because 3x means 3 groups of x, and 5x means 5 groups of x, together they make 8 groups of x.
合并同类项实际上是乘法分配律的应用。3x 表示 3 个 x，5x 表示 5 个 x，合在一起就是 8 个 x。
3x + 5x = (3 + 5)x = 8x
Similarly, subtraction works the same way:
减法也同样适用：
9y – 4y = (9 – 4)y = 5y
This shows that only the coefficients are added or subtracted; the variable part remains unchanged.
这说明合并时只对系数进行加减，字母部分保持不变。
4. Combining Positive and Negative Coefficients | 合并正负系数
When combining terms with signs, think of each term as carrying its own sign. For example, -3x + 2x means subtracting 3x then adding 2x, which gives -x.
合并带符号的项时，要把每一项前面的正负号看作它自己的符号。例如，-3x + 2x 表示先减去 3x 再加上 2x，结果是 -x。
-3x + 2x = -1x = -x
Another example is 7m – 3m + 2m:
另一个例子是 7m – 3m + 2m：
7m – 3m + 2m = (7 – 3 + 2)m = 6m
Always keep the sign with the term before combining. A common mistake is to ignore the negative sign in front of a term.
合并前务必保留每一项前面的符号。一个常见错误是忽略项前面的负号。
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