Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain several terms. Some of these terms have exactly the same variable part, and they are called ‘like terms’. Collecting like terms means adding or subtracting these terms to make the expression shorter and simpler.

在代数中，表达式通常包含若干项。其中一些项具有完全相同的变量部分，它们被称为“同类项”。合并同类项就是把这些项相加或相减，使表达式更短、更简洁。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have the same variables raised to the same powers. The coefficients (the numbers in front) do not need to be the same.

同类项是指拥有相同变量且相同指数的项。系数（即变量前的数字）不一定相同。

For example, 3x and 5x are like terms because both have the variable x. Similarly, 7ab and 2ab are like terms because both have the variable part ab.

例如，3x 和 5x 是同类项，因为它们都有变量 x。同样，7ab 和 2ab 是同类项，因为它们都有变量部分 ab。

However, 2x and 2y are not like terms because the variables are different. Also, 4x² and 4x are not like terms because the powers of x are different.

然而，2x 和 2y 不是同类项，因为变量不同。此外，4x² 和 4x 也不是同类项，因为 x 的指数不同。

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

Collecting like terms simplifies an expression. A shorter expression is easier to read, easier to evaluate, and easier to use in equations.

合并同类项可以化简表达式。更简短的表达式更容易阅读、更容易代入求值，也更容易在方程中使用。

For instance, 2x + 3x + 4 can be simplified to 5x + 4. This takes up less space and immediately shows the structure of the expression.

例如，2x + 3x + 4 可以化简为 5x + 4。这样占用空间更小，也能立即看出表达式的结构。

It also helps when solving equations. If an equation contains many like terms, combining them first makes the next step clearer.

在解方程时也很有帮助。如果方程中包含许多同类项，先合并它们会使后续步骤更清晰。

3. Identifying Like Terms | 识别同类项

To identify like terms, look at the variable part of each term. Ignore the coefficient for a moment and check whether the variables and exponents are identical.

要识别同类项，需要观察每一项的变量部分。暂时忽略系数，检查变量和指数是否完全相同。

Here are some examples:

以下是一些例子：

5a and 12a are like terms because both contain a.

5a 和 12a 是同类项，因为两者都含有 a。
3m²n and 8m²n are like terms because both contain m²n.

3m²n 和 8m²n 是同类项，因为两者都含有 m²n。
6xy and 6yx are like terms because xy is the same as yx (multiplication is commutative).

6xy 和 6yx 是同类项，因为 xy 与 yx 相同（乘法满足交换律）。
2p³ and 2p² are not like terms because the exponents of p are different.

2p³ 和 2p² 不是同类项，因为 p 的指数不同。

4. The Distributive Law and Combining | 分配律与合并

Combining like terms is based on the distributive law. For example, 3x + 5x means you have 3 lots of x plus 5 lots of x, which gives 8 lots of x.

合并同类项基于分配律。例如，3x + 5x 表示 3 个 x 加上 5 个 x，一共得到 8 个 x。

3x + 5x = (3 + 5)x = 8x

So we add the coefficients, 3 and 5, and keep the variable x unchanged.

因此我们将系数 3 和 5 相加，并保持变量 x 不变。

The same rule works for any like terms. If the expression is 4a – 2a, then 4 – 2 = 2, so the result is 2a.

同样的规则适用于任何同类项。如果表达式是 4a – 2a，那么 4 – 2 = 2，结果是 2a。

5. Combining Terms with Coefficients | 合并带系数的项

When you combine like terms, you only add or subtract the coefficients. The variable part remains exactly the same.

合并同类项时，只对系数进行加减。变量部分保持不变。

For example, 7b + 2b – 3b can be simplified by first adding: 7 + 2 – 3 = 6. Therefore 7b + 2b – 3b = 6b.

例如，7b + 2b – 3b 可以先计算系数：7 + 2 – 3 = 6。因此 7b + 2b – 3b = 6b。

7b + 2b – 3b = (7 + 2 – 3)b = 6b

Always remember: never add the exponents when combining like terms. Only the coefficients change.

始终记住：合并同类项时绝不能把指数相加。只有系数发生变化。

6. Handling Constants | 处理常数项

Constants are numbers without variables, such as 5, -3, or 0.5. All constants are like terms with each other.

常数项是不含变量的数字，如 5、-3 或 0.5。所有常数项彼此都是同类项。

So 4 + 7 – 2 can be simplified to 9. You can combine all constant terms in an expression just like you combine variable terms.

因此 4 + 7 – 2 可以化简为 9。你可以像合并变量项一样合并表达式中的所有常数项。

For example, 2x + 5 + 3x – 1 first has like variable terms 2x and 3x, and constant terms 5 and -1. The simplified form is 5x + 4.

例如，2x + 5 + 3x – 1 中，变量同类项是 2x 和 3x，常数项是 5 和 -1。化简后为 5x + 4。

7. Combining Multiple Variables | 多个变量的合并

When an expression contains different variable combinations, you must group only the identical combinations.

当表达式包含不同的变量组合时，必须只将完全相同的组合归为一组。

Take 3xy + 2x + 5xy – y. The terms 3xy and 5xy are like terms; 2x and -y each stand alone because no other term has the same variable part.

以 3xy + 2x + 5xy – y 为例。3xy 和 5xy 是同类项；2x 和 -y 各自独立，因为没有其他项具有相同的变量部分。

3xy + 2x + 5xy – y = (3 + 5)xy + 2x – y = 8xy + 2x – y

Notice that 2x and -y remain untouched because they are not like terms with xy terms.

注意 2x 和 -y 保持不变，因为它们与 xy 项不是同类项。

8. Negative Coefficients and Subtraction | 负系数与减法

Subtraction must be handled carefully. The sign in front of a term belongs to that coefficient.

减法必须小心处理。项前面的符号属于该项的系数。

For example, -2y + 7y is the same as (-2 + 7)y, which equals 5y.

例如，-2y + 7y 等价于 (-2 + 7)y，结果为 5y。

Similarly, 3a – 5a gives (-2)a, usually written as -2a.

同样，3a – 5a 得到 (-2)a，通常写作 -2a。

3a – 5a = (3 – 5)a = -2a

If the coefficient is 1 or -1, it is often omitted when written at the start of a term. For instance, 1p is written as p, and -1q is written as -q.

当系数为 1 或 -1 时，通常省略不写。例如，1p 写作 p，-1q 写作 -q。

9. Squares and Higher Powers | 平方与高次幂

Like terms must have the same exponent on each variable. This applies to squares, cubes, and higher powers.

同类项要求每个变量的指数都相同。这适用于平方、立方以及更高次幂。

For example, 4x² and 9x² are like terms, so 4x² + 9x² = 13x². But 4x² and 9x are not like terms because the exponent of x is different.

例如，4x² 和 9x² 是同类项，所以 4x² + 9x² = 13x²。但 4x² 和 9x 不是同类项，因为 x 的指数不同。

Similarly, 2p³q and 5p³q are like terms, so they combine to 7p³q. The term 2p²q cannot be combined with them.

同样，2p³q 和 5p³q 是同类项，因此合并为 7p³q。而 2p²q 不能与它们合并。

10. Simplifying Long Expressions | 化简长表达式

For longer expressions, it helps to rearrange terms so that like terms are next to each other. Then combine each group one by one.

对于较长的表达式，可以重新排列各项，使同类项相邻。然后再逐组合并。

Consider 8x – 3 + 2y – 5x + 7 – y. Group the variable terms and constants:

考虑 8x – 3 + 2y – 5x + 7 – y。将变量项和常数项分组：

x terms: 8x – 5x = 3x

x 项：8x – 5x = 3x
y terms: 2y – y = 1y = y

y 项：2y – y = 1y = y
Constants: -3 + 7 = 4

常数项：-3 + 7 = 4

The simplified expression is 3x + y + 4.

化简后的表达式为 3x + y + 4。

Notice that the order of terms in an addition expression can be changed. We usually write variable terms in alphabetical order or by descending powers, but this is not strictly required.

注意，在加法表达式中，项的顺序可以改变。我们通常按字母顺序或按降幂排列变量项，但这并不是严格必需的。

11. Common Mistakes to Avoid | 常见错误

Students often make a few predictable mistakes when collecting like terms. Watch out for these:

学生在合并同类项时往往会犯一些可预测的错误。请注意以下几点：

Combining terms that are not like terms, such as 3x + 2y. They cannot be simplified further.

合并不是同类项的项，例如 3x + 2y。它们不能再被化简。
Adding exponents when combining coefficients, such as writing 3x + 5x = 8x². This is wrong; the correct result is 8x.

在合并系数时把指数也相加，例如写成 3x + 5x = 8x²。这是错误的，正确结果是 8x。
Forgetting the negative sign in front of a term, for example treating 7 – 3x + 2x as 7 – 5x instead of 7 – x.

忽略项前面的负号，例如将 7 – 3x + 2x 误算为 7 – 5x，而正确结果是 7 – x。
Thinking that xy and y² are like terms. They are not, because the variables are different.

认为 xy 和 y² 是同类项。它们不是，因为变量不同。

Always check each term carefully before combining.

合并前务必仔细检查每一项。

12. Practice Problems with Solutions | 练习与解答

Try these examples yourself, then check the answers below.

请先自行尝试以下例题，再核对下方答案。

Problem 1: Simplify 6a + 3a – 2a

练习1：化简 6a + 3a – 2a
Problem 2: Simplify 5x + 7 – 2x + 3

练习2：化简 5x + 7 – 2x + 3
Problem 3: Simplify 4m² + 3n – m² + 5n

练习3：化简 4m² + 3n – m² + 5n
Problem 4: Simplify 2ab – ab + 7 – 3ab + 4

练习4：化简 2ab – ab + 7 – 3ab + 4

Solutions:

解答：

Problem 1: 6a + 3a – 2a = (6 + 3 – 2)a = 7a

练习1：6a + 3a – 2a = (6 + 3 – 2)a = 7a
Problem 2: 5x – 2x + 7 + 3 = 3x + 10

练习2：5x – 2x + 7 + 3 = 3x + 10
Problem 3: 4m² – m² + 3n + 5n = 3m² + 8n

练习3：4m² – m² + 3n + 5n = 3m² + 8n
Problem 4: 2ab – ab – 3ab + 7 + 4 = (2 – 1 – 3)ab + 11 = -2ab + 11

练习4：2ab – ab – 3ab + 7 + 4 = (2 – 1 – 3)ab + 11 = -2ab + 11

Collecting like terms is a fundamental skill in algebra. Once you master it, simplifying expressions becomes quick and reliable.

合并同类项是代数学习中的基本技能。一旦掌握，化简表达式就会变得快速而可靠。

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Collecting Like Terms | 合并同类项