📚 Collecting Like Terms | 合并同类项
Algebra is the language of mathematics, and one of its most fundamental skills is collecting like terms. This process allows us to simplify expressions by grouping terms that share the same variable part, making equations easier to understand and solve.
代数是数学的语言，而合并同类项是最基础的技能之一。通过将具有相同变量部分的项分组，我们可以化简表达式，使方程更易于理解与求解。
1. What Are Like Terms? | 什么是同类项
Like terms are terms that have exactly the same variable part. The variable part includes both the letters and their exponents. For example, 3x and 5x are like terms because both contain the single variable x. Similarly, 2y² and -7y² are like terms because both contain y².
同类项是指变量部分完全相同的项。变量部分包括字母及其指数。例如，3x 和 5x 是同类项，因为两者都只含有变量 x。同样，2y² 和 -7y² 也是同类项，因为两者都含有 y²。
- 4a and 9a are like terms (both contain a).
- 4a 和 9a 是同类项（都含有 a）。
- 5xy and -2xy are like terms (both contain xy).
- 5xy 和 -2xy 是同类项（都含有 xy）。
- 3x² and 8x² are like terms (both contain x²).
- 3x² 和 8x² 是同类项（都含有 x²）。
2. Unlike Terms | 异类项
Unlike terms have different variable parts, either because the letters differ or because the exponents differ. For instance, 4x and 4x² are not like terms, since the power of x is different. Likewise, 3a and 3b are unlike terms because they involve different letters.
异类项的变量部分不同，可能是字母不同，也可能是指数不同。例如，4x 和 4x² 不是同类项，因为 x 的幂不同。同样，3a 和 3b 是异类项，因为它们涉及的字母不同。
- 2x and 3y are unlike terms (different letters).
- 2x 和 3y 是异类项（字母不同）。
- 5p and 5p³ are unlike terms (different exponents).
- 5p 和 5p³ 是异类项（指数不同）。
- 6mn and 6m are unlike terms (one has mn, the other only m).
- 6mn 和 6m 是异类项（一个含 mn，另一个只含 m）。
3. The Coefficient Rule | 系数规则
ficients, while keeping the variable part unchanged. For example, 7x + 2x = (7 + 2)x = 9x.
系数是乘以变量部分的数字因子。合并同类项时，我们只对系数进行加减运算，而保持变量部分不变。例如，7x + 2x = (7 + 2)x = 9x。
ax + bx = (a + b)x
applies to any variable part, including powers. So 5y³ + 3y³ = 8y³, and -2ab + 4ab = 2ab.
此规则适用于任何变量部分，包括幂。所以 5y³ + 3y³ = 8y³，而 -2ab + 4ab = 2ab。
4. Steps for Collecting Like Terms | 合并同类项的步骤
To collect like terms systematically, follow these steps:
要系统地合并同类项，请按以下步骤操作：
- Identify all like terms in the expression.
- 识别表达式中所有的同类项。
- Group them together, either mentally or by underlining.
- 将它们分组，可在心里分组或画下划线标记。
- Add or subtract the coefficients of each group.
- 对每个组的系数进行加或减运算。
- Write the simplified expression with the variable part unchanged.
- 写出变量部分保持不变的化简式。
5. Worked Examples | 实例解析
contain a, so they are like terms. Adding their coefficients: 3 + 2 + 5 = 10, giving 10a.
让我们化简 3a + 2a + 5a。三个项都含有 a，因此它们是同类项。将系数相加：3 + 2 + 5 = 10，得到 10a。
Now consider 4x + 2y - x + 5y. Group like terms: (4x - x) + (2y + 5y). Simplifying gives 3x + 7y.
现在考虑 4x + 2y - x + 5y。将同类项分组：(4x - x) + (2y + 5y)。化简得到 3x + 7y。
- m. Group m terms: 7m + 2m - m = 8m. Group n terms: -3n + 6n = 3n. The result is 8m + 3n.
一个更长的例子：7m - 3n + 2m + 6n - m。将 m 项分组：7m + 2m - m = 8m。将 n 项分组：-3n + 6n = 3n。结果为 8m + 3n。
6. Expressions with Mixed Powers | 含混合幂的表达式
When an expression contains terms with different powers, each power must be treated separately. For example, in the expression 2x² + 3x + 4x² - x, the x² terms combine to 6x², while the x terms combine to 2x. The final answer is 6x² + 2x.
当表达式包含不同幂的项时，每个幂必须单独处理。例如，在表达式 2x² + 3x + 4x² - x 中，x² 项合并为 6x²，而 x 项合并为 2x。最终答案为 6x² + 2x。
2x² + 3x + 4x² - x = 6x² + 2x
Remember that x² and x are not like terms, so they cannot be combined into a single term.
记住 x² 和 x 不是同类项，因此不能合并成一个项。
7. Common Mistakes | 常见错误
2x = 3x, not simply 3x (which is correct only if the first term is written clearly as 1x).
一个常见错误是将 x 与 x² 合并。由于它们的变量部分不同，必须保持分开。另一个错误是忘记系数 1，例如 x + 2x = 3x，只有当第一个项明确写为 1x 时才正确。
- Incorrect: 5x² + 2x = 7x² or 7x.
- 错误：5x² + 2x = 7x² 或 7x。
- Incorrect: 3ab + 2a = 5ab.
- 错误：3ab + 2a = 5ab。
- Correct: 5x² + 2x stays as it is.
- 正确：5x² + 2x 保持不变。
8. Collecting Like Terms with Brackets | 含括号的合并
rms: 6x + 6x = 12x. Collecting constants: 8 - 3 = 5. The simplified expression is 12x + 5.
当出现括号时，先用分配律展开括号，再合并同类项。考虑 2(3x + 4) + 3(2x - 1)。展开得到 6x + 8 + 6x - 3。合并 x 项：6x + 6x = 12x。合并常数项：8 - 3 = 5。化简后的表达式为 12x + 5。
2(3x + 4) + 3(2x - 1) = 12x + 5
Always check whether the original expression contains subtraction or negative signs, as they affect the coefficients during distribution.
始终检查原表达式是否包含减法或负号，因为它们会影响展开时的系数。
9. Applications in Geometry | 在几何中的应用
3x - 1 is calculated as 2(length + width) = 2[(4x + 2) + (3x - 1)] = 2(7x + 1) = 14x + 2.
合并同类项常出现在周长和面积问题中。例如，长为 4x + 2、宽为 3x - 1 的矩形，其周长为 2(长 + 宽) = 2[(4x + 2) + (3x - 1)] = 2(7x + 1) = 14x + 2。
Similarly, the sum of two polynomials often requires collecting like terms, such as adding (3x² + 2x - 5) and (x² - 4x + 7) to get 4x² - 2x + 2.
类似地，两个多项式的相加通常需要合并同类项，例如将 (3x² + 2x - 5) 与 (x² - 4x + 7) 相加，得到 4x² - 2x + 2。
10. Practice Questions | 练习题目
Test your understanding with the following questions:
通过以下题目测试你的理解：
- Simplify: 6a + 3a - 2a.
- 化简：6a + 3a - 2a。
- Simplify: 5b² + 2b - b² + 4b.
- 化简：5b² + 2b - b² + 4b。
- Simplify: 4(2x + 3) - 2(x - 1).
- 化简：4(2x + 3) - 2(x - 1)。
- Simplify: 3pq + 2p - pq + 5p.
- 化简：3pq + 2p - pq + 5p。
Answers: 1) 7a; 2) 4b² + 6b; 3) 6x + 14; 4) 2pq + 7p.
答案：1) 7a；2) 4b² + 6b；3) 6x + 14；4) 2pq + 7p。
11. Summary | 总结
Collecting like terms is a cornerstone of algebra. By identifying terms with identical variable parts, combining their coefficients, and preserving the variable part, you can simplify even complex expressions. Practice with both simple and bracket-inclusive expressions to build confidence.
合并同类项是代数的基石。通过识别具有相同变量部分的项、合并它们的系数并保留变量部分，你可以化简甚至复杂的表达式。通过练习简单表达式和含括号的表达式来建立自信。
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