Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Collecting like terms is a fundamental skill in algebra. It means simplifying an expression by adding or subtracting terms that have exactly the same variable parts. This process makes expressions shorter, easier to read, and ready for further manipulation.

合并同类项是代数中的一项基本技能。它指的是将含有完全相同变量部分的项通过加法或减法进行合并，从而使表达式变得更简洁、易于阅读，并为后续的运算做好准备。

1. What Are Like Terms? | 什么是同类项？

A term is a number, a variable, or a product of numbers and variables. For example, 3x, -5y, 7, and 2xy are all terms. Like terms are terms that have the exact same variable part, including the same powers of variables. Only the numerical coefficients may be different.

项是一个数、一个变量，或者数与变量的乘积。例如：3x、-5y、7 和 2xy 都是项。同类项是指变量部分完全相同的项，包括变量的指数也相同。只有数字系数可以不同。

For instance, 4x and 9x are like terms because both contain the same variable x. However, 4x and 4x² are not like terms, because one has x and the other has x². Constants, such as 5 and -3, are also like terms with each other.

例如，4x 和 9x 是同类项，因为它们都含有相同的变量 x。但是 4x 和 4x² 不是同类项，因为一个含 x，另一个含 x²。常数项，如 5 和 -3，也是彼此相同的同类项。

Examples: 7a and 3a → like terms; 7a and 3b → unlike terms; 2xy and 5xy → like terms.

例：7a 与 3a → 同类项；7a 与 3b → 非同类项；2xy 与 5xy → 同类项。

2. Adding and Subtracting Like Terms | 同类项的加减

To collect like terms, you add or subtract their coefficients. Keep the variable part unchanged. For example, 3x + 5x = (3 + 5)x = 8x. In the same way, 12y – 7y = (12 – 7)y = 5y.

合并同类项时，只需要对系数进行加法或减法，变量部分保持不变。例如：3x + 5x = (3 + 5)x = 8x。同理，12y – 7y = (12 – 7)y = 5y。

If there are several terms, group the like terms first. For example, simplify 6a + 2b – 3a + 4b. Group the a terms together and the b terms together: (6a – 3a) + (2b + 4b) = 3a + 6b.

如果表达式中有多个项，应先将同类项分组。例如，化简 6a + 2b – 3a + 4b。将含 a 的项和含 b 的项分别归组：(6a – 3a) + (2b + 4b) = 3a + 6b。

TIP: Always check the sign in front of each term. The sign belongs to the coefficient that follows it.
提示： 始终注意每项前面的符号。符号属于其后的系数。

3. When Terms Cannot Be Combined | 何时不能合并

Unlike terms cannot be combined. In algebra, you cannot add a term with x and a term with y in the same way you cannot add apples and oranges. The expression 2x + 3y must remain as 2x + 3y.

非同类项不能合并。在代数中，不能直接将含 x 的项与含 y 的项相加，就像不能把苹果和橘子直接相加一样。表达式 2x + 3y 必须保持为 2x + 3y。

Similarly, x² and x cannot be added. They are different powers of the same variable. For example, 5x² + 2x – 3x² + 4x can be simplified to (5x² – 3x²) + (2x + 4x) = 2x² + 6x, but x² and x are never combined directly.

同样，x² 与 x 也不能相加。它们是同一变量的不同幂次。例如，5x² + 2x – 3x² + 4x 可化简为 (5x² – 3x²) + (2x + 4x) = 2x² + 6x，但 x² 与 x 永远不会直接合并。

4. Using the Distributive Property | 应用乘法分配律

Often, an expression contains brackets. Use the distributive property to remove brackets before collecting like terms. The distributive property states that a(b + c) = ab + ac.

表达式中常常含有括号。在合并同类项之前，需要用乘法分配律去掉括号。乘法分配律为：a(b + c) = ab + ac。

For example, simplify 3(x + 2) + 2(x – 1). First expand: 3x + 6 + 2x – 2. Then collect like terms: (3x + 2x) + (6 – 2) = 5x + 4.

例如，化简 3(x + 2) + 2(x – 1)。先展开：3x + 6 + 2x – 2。再合并同类项：(3x + 2x) + (6 – 2) = 5x + 4。

3(x + 2) + 2(x – 1) = 3x + 6 + 2x – 2 = 5x + 4

Be careful with negative signs outside brackets. For instance, -2(x – 3) means -2 × x + (-2) × (-3) = -2x + 6. Many mistakes happen by forgetting to multiply every term inside the bracket.

要特别注意括号外的负号。例如，-2(x – 3) 表示 -2 × x + (-2) × (-3) = -2x + 6。许多错误都源于忘记将括号内的每一项都乘以括号外的数。

5. Collecting Terms with Indices (Powers) | 含指数的同类项合并

Like terms must have the same variable raised to the same power. Thus 2x³ and 5x³ are like terms, but 2x³ and 5x² are not. When adding or subtracting, keep the power unchanged and combine only the coefficients.

同类项必须含有相同指数下的相同变量。因此 2x³ 与 5x³ 是同类项，但 2x³ 与 5x² 不是。进行加减时，保持指数不变，只合并系数。

For example, 4x³ + x² – 2x³ + 3x² = (4 – 2)x³ + (1 + 3)x² = 2x³ + 4x². Note that 4x³ and 3x² are not collected together.

例如，4x³ + x² – 2x³ + 3x² = (4 – 2)x³ + (1 + 3)x² = 2x³ + 4x²。注意 4x³ 与 3x² 不合并。

Expression	Simplify	Reason
2x² + 3x²	5x²	Same variable x²
2x² + 3x	2x² + 3x	Different powers
5x⁴ – x⁴	4x⁴	Same variable x⁴

6. Negative and Fractional Coefficients | 负系数与分数系数

Negative coefficients follow the same rules. For example, -2x + 5x = 3x, because -2 + 5 = 3. Also, -3y – 4y = -7y, because -3 – 4 = -7.

负系数的运算规则完全相同。例如，-2x + 5x = 3x，因为 -2 + 5 = 3。同样，-3y – 4y = -7y，因为 -3 – 4 = -7。

Fractional coefficients can be combined using fraction arithmetic. For example, ½x + ⅓x = (½ + ⅓)x = ⁵⁄₆x. Remember to convert to a common denominator when adding fractions.

分数系数可通过分数运算进行合并。例如，½x + ⅓x = (½ + ⅓)x = ⁵⁄₆x。注意进行分数加法时需要通分。

Let’s simplify ¾p – ½p. Rewrite ½ as ²⁄₄, so ¾p – ²⁄₄p = (¾ – ²⁄₄)p = ¼p. The variable part p stays unchanged.

化简 ¾p – ½p：将 ½ 改写为 ²⁄₄，则 ¾p – ²⁄₄p = (¾ – ²⁄₄)p = ¼p。变量部分 p 保持不变。

7. Combining Terms Involving Products and Multiple Variables | 含乘积与多变量的项合并

Terms such as 4xy and 7xy are like terms because both contain xy. The order of variables in a term does not matter: xy and yx are the same. For example, 3ab + 2ba = 3ab + 2ab = 5ab.

像 4xy 和 7xy 这样的项是同类项，因为它们都含有 xy。项中变量的顺序无关紧要：xy 与 yx 相同。例如，3ab + 2ba = 3ab + 2ab = 5ab。

However, xy and x²y are not like terms, because one has x times y and the other has x² times y. Also, xy and xz are unlike because they contain different variable combinations.

但是，xy 与 x²y 不是同类项，因为前者是 x×y，后者是 x²×y。此外，xy 与 xz 也不是同类项，因为它们含有的变量组合不同。

Consider the expression 2mn + 3n – 4mn + 2n. Group the mn terms and the n terms: (2mn – 4mn) + (3n + 2n) = -2mn + 5n.

考虑表达式 2mn + 3n – 4mn + 2n。合并含 mn 的项和含 n 的项：(2mn – 4mn) + (3n + 2n) = -2mn + 5n。

8. Applying like Terms in Word Problems | 在应用题中应用合并同类项

Collecting like terms is useful when modelling real-world situations. Suppose a rectangle has length (3x + 4) cm and width (2x – 1) cm. The perimeter is 2 × (length + width) = 2 × [(3x + 4) + (2x – 1)] = 2 × (5x + 3) = 10x + 6 cm.

合并同类项在建立现实情境模型时非常有用。假设一个长方形的长为 (3x + 4) cm，宽为 (2x – 1) cm。其周长为 2 × (长 + 宽) = 2 × [(3x + 4) + (2x – 1)] = 2 × (5x + 3) = 10x + 6 cm。

For example, if x = 5, the perimeter is 10 × 5 + 6 = 56 cm. Simplifying first makes substitution easier and less error-prone.

例如，若 x = 5，则周长为 10 × 5 + 6 = 56 cm。先化简再代入会使计算更简单，也不易出错。

Another example: Tom has 5 boxes with x apples each, and 3 extra apples. His friend gives him 2 more boxes of x apples and takes away 1 apple. Total apples: 5x + 3 + 2x – 1 = 7x + 2.

另一个例子：汤姆有 5 盒苹果，每盒 x 个，另外还有 3 个苹果。他的朋友又给了他 2 盒（每盒 x 个），并拿走了 1 个苹果。苹果总数为：5x + 3 + 2x – 1 = 7x + 2。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One common mistake is combining unlike terms, such as writing 2x + 3y = 5xy. This is incorrect. Another is accidentally changing the variable power, e.g., x + x² = x³ is wrong; they are not like terms.

一个常见错误是合并非同类项，例如写出 2x + 3y = 5xy。这是不正确的。另一个错误是意外地改变了变量的指数，比如 x + x² = x³ 是错误的；它们不是同类项。

Students often drop negative signs when moving terms. For example, in 5x – 2 – 3x + 7, the -2 and +7 are constants. The correct simplification is (5x – 3x) + (-2 + 7) = 2x + 5. Keep the subtraction sign with the 2.

学生在移动项时经常丢失负号。例如，在 5x – 2 – 3x + 7 中，-2 和 +7 是常数项。正确的化简是 (5x – 3x) + (-2 + 7) = 2x + 5。要把减号和 2 一起保留。

✓ Strategy | 解题策略

Underline like terms with the same pattern before adding.
在相加前用相同标记标出同类项。
Rewrite the expression by grouping like terms together.
重新书写表达式，将同类项分组。
Simplify each group separately, then write the final answer.
分别化简每组，然后写出最终答案。

10. Practice Questions | 练习题目

Try these questions on your own. You can check the answers after each one.

请独立尝试以下题目。每道题后可以核对答案。

Q1: Simplify 7a + 3a – 2a.
Q1：化简 7a + 3a – 2a。

Answer: 8a

答案：8a

Q2: Simplify 5x² + 3x – 2x² + 4x.
Q2：化简 5x² + 3x – 2x² + 4x。

Answer: 3x² + 7x

答案：3x² + 7x

Q3: Simplify 2(3t + 1) – (t – 2).
Q3：化简 2(3t + 1) – (t – 2)。

Answer: 6t + 2 – t + 2 = 5t + 4

答案：6t + 2 – t + 2 = 5t + 4

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Collecting Like Terms | 合并同类项