📚 Collecting Like Terms | 合并同类项
In algebra, "collecting like terms" is the process of simplifying an expression by adding or subtracting terms that have the same variable part. This is one of the most fundamental skills you need for IGCSE Mathematics, as it appears in almost every algebraic topic.
在代数中，“合并同类项”是指将含有相同变量部分的项进行加减，从而简化表达式的过程。这是 IGCSE 数学中最基础的技能之一，几乎出现在所有代数专题中。
1. What Is a Term? | 什么是“项”？
A term is a single number, a variable, or a product of numbers and variables. For example, in the expression 3x + 5y - 2, the terms are 3x, 5y, and -2. Terms are separated by plus or minus signs.
“项”是一个单独的数字、一个变量，或数字与变量的乘积。例如，在表达式 3x + 5y − 2 中，项为 3x、5y 和 −2。项与项之间用加号或减号分隔。
Each term has two parts: a coefficient (the number in front) and a variable part. In the term 7x2, the coefficient is 7 and the variable part is x2.
每一项都由两部分组成：系数（前面的数字）和变量部分。在项 7x² 中，系数是 7，变量部分是 x²。
2. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable part, including the same exponents. For example, 4x and -9x are like terms because both contain x1. Similarly, 2x2y and 5x2y are like terms because both contain x2y.
同类项是指变量部分完全相同（包括指数相同）的项。例如，4x 和 −9x 是同类项，因为它们都含有 x¹。同样，2x²y 和 5x²y 是同类项，因为它们都含有 x²y。
Constant terms are also like terms with each other. For instance, 3 and −8 are like terms.
常数项之间也互为同类项。例如，3 和 −8 是同类项。
3. The Rule for Collecting Like Terms | 合并同类项的规则
Only like terms can be combined. To combine them, add or subtract their coefficients while keeping the variable part unchanged.
只有同类项才能合并。合并时，将它们的系数进行加减，变量部分保持不变。
ax + bx = (a + b)x
For example, 5x + 3x = 8x. The variable x does not change; we simply add the coefficients 5 and 3.
例如，5x + 3x = 8x。变量 x 保持不变，我们只需将系数 5 和 3 相加。
4. Collecting Terms with Addition | 合并加法中的同类项
Suppose you have the expression 2a + 7a + 4a. All three terms are like terms because they all contain a. Add the coefficients: 2 + 7 + 4 = 13. So the simplified expression is 13a.
假设有表达式 2a + 7a + 4a。这三项都是同类项，因为它们都含有 a。将系数相加：2 + 7 + 4 = 13。因此化简后的表达式是 13a。
Let's try a more complex example: 6m + 3n + 2m + n. First, identify the like terms: 6m and 2m are like; 3n and n are like. Combine each pair: 6m + 2m = 8m, and 3n + n = 4n. The final expression is 8m + 4n.
让我们看一个更复杂的例子：6m + 3n + 2m + n。首先识别同类项：6m 和 2m 是同类；3n 和 n 是同类。分别合并：6m + 2m = 8m，3n + n = 4n。最终表达式为 8m + 4n。
5. Collecting Terms with Subtraction | 合并减法中的同类项
When subtracting like terms, be careful with negative signs. The expression 9p - 4p - 2p can be simplified by combining coefficients: 9 - 4 - 2 = 3, so the result is 3p.
当合并减法中的同类项时，要特别小心负号。表达式 9p − 4p − 2p 可以通过合并系数来简化：9 − 4 − 2 = 3，所以结果是 3p。
Consider 5x2 - 3x2 + 2x2. All terms contain x2, so combine coefficients: 5 - 3 + 2 = 4. The answer is 4x2.
考虑 5x² − 3x² + 2x²。所有项都含有 x²，所以合并系数：5 − 3 + 2 = 4。答案是 4x²。
6. Combining Different Variables | 合并不同变量
Terms with different variables cannot be combined. For example, 4x + 2y cannot be simplified further because x and y are different variables.
不同变量的项不能合并。例如，4x + 2y 不能再化简，因为 x 和 y 是不同的变量。
Even if the variables look similar, they must have exactly the same variable part. 3ab and 4a are not like terms because one has ab and the other only has a.
即使变量看起来相似，也必须具有完全相同的变量部分。3ab 和 4a 不是同类项，因为一个含有 ab，另一个只含有 a。
7. Handling Constants | 处理常数项
Constant terms are like terms with each other. For example, in 3x + 5 + 2x - 1, the constants are 5 and −1. Combine them: 5 - 1 = 4. The variable terms 3x and 2x combine to 5x. So the simplified expression is 5x + 4.
常数项之间互为同类项。例如，在 3x + 5 + 2x − 1 中，常数项是 5 和 −1。合并：5 − 1 = 4。变量项 3x 和 2x 合并为 5x。因此化简后的表达式是 5x + 4。
Always keep the constant term separate from variable terms. In the final answer, it is conventional to write the variable terms first, followed by the constant term.
始终将常数项与变量项分开。在最终答案中，通常先写变量项，再写常数项。
8. Collecting Terms with Exponents | 合并含指数的同类项
When terms have exponents, the exponents must match exactly for the terms to be like terms. For instance, 4x2 and 7x2 are like terms, but 4x2 and 7x are not.
当项含有指数时，指数必须完全相同，这些项才是同类项。例如，4x² 和 7x² 是同类项，但 4x² 和 7x 不是。
Consider 2x3 + 5x2 - x3 + 3x2. Group the like terms: 2x3 - x3 = x3, and 5x2 + 3x2 = 8x2. The result is x3 + 8x2. Notice we never combine x3 with x2.
考虑 2x³ + 5x² − x³ + 3x²。将同类项分组：2x³ − x³ = x³，5x² + 3x² = 8x²。结果是 x³ + 8x²。注意我们绝不将 x³ 与 x² 合并。
9. Collecting Terms with Fractions and Decimals | 合并含分数与小数的同类项
Fractions and decimals follow the same rules. For example, 12x + 32x = 42x = 2x.
分数和小数遵循同样的规则。例如，½x + 3⁄2x = 4⁄2x = 2x。
With decimals: 0.3x2 + 0.7x2 = 1.0x2 = x2. Just add the coefficients as you normally would, then keep the variable part unchanged.
对于小数：0.3x² + 0.7x² = 1.0x² = x²。像平常一样将系数相加，然后保持变量部分不变。
2.5y + 1.5y = 4y
10. Common Mistakes to Avoid | 常见的错误
One common mistake is trying to combine terms that are not like terms, such as 3x + 2x2 = 5x3. This is incorrect. Always check that the variable parts match exactly.
一个常见错误是试图合并不是同类项的项，例如 3x + 2x² = 5x³。这是错误的。始终检查变量部分是否完全相同。
Another mistake is forgetting the sign before a term. When collecting terms, treat the sign as part of the coefficient. For example, in 2x - 5x + 3x, the coefficients are 2, −5, and 3.
另一个错误是忘记项前面的符号。在合并时，将符号视为系数的一部分。例如，在 2x − 5x + 3x 中，系数是 2、−5 和 3。
Also, remember that x means 1x, so x + x = 2x, not x2.
另外，请记住 x 就是 1x，所以 x + x = 2x，而不是 x²。
11. Real-World Application | 实际应用
Collecting like terms is used in solving equations, simplifying formulas, and even in real-life calculations. For example, if you buy 3 pens and then 2 more pens, the total is 3p + 2p = 5p, where p represents the price of one pen.
合并同类项用于解方程、简化公式，甚至用于实际生活中的计算。例如，如果你买了 3 支笔，又买了 2 支笔，总共就是 3p + 2p = 5p，其中 p 代表一支笔的价格。
In geometry, you might need to combine like terms when finding the perimeter of a shape. If a rectangle has sides 2x + 3 and x - 1, its perimeter is 2(2x + 3) + 2(x - 1) = 4x + 6 + 2x - 2 = 6x + 4.
在几何中，求图形周长时可能需要合并同类项。如果一个矩形的两边分别为 2x + 3 和 x − 1，其周长为 2(2x + 3) + 2(x − 1) = 4x + 6 + 2x − 2 = 6x + 4。
12. Practice Questions | 练习题目
Try these questions to test your understanding.
尝试以下题目来测试你的理解。
Simplify 5x + 3x + 2x.
化简 5x + 3x + 2x。
Simplify 7a - 2a + 4b - b.
化简 7a − 2a + 4b − b。
Simplify 3x2 + 2x - x2 + 5x.
化简 3x² + 2x − x² + 5x。
Simplify 0.4m + 1.2m - 0.8m + 6.
化简 0.4m + 1.2m − 0.8m + 6。
Answers: 10x; 5a + 3b; 2x2 + 7x; 0.8m + 6.
答案：10x；5a + 3b；2x² + 7x；0.8m + 6。
13. Summary | 总结
Collecting like terms is a simple but essential skill. Always identify terms with the same variable part, combine their coefficients, and keep the variable part unchanged. Never mix different variables or different exponents.
合并同类项是一项简单但至关重要的技能。始终识别具有相同变量部分的项，合并它们的系数，并保持变量部分不变。绝不混合不同的变量或不同的指数。
With practice, you will be able to simplify even complex expressions quickly and accurately. Keep practising and check your signs carefully!
通过练习，你将能够快速准确地化简复杂的表达式。继续练习，并仔细检查你的符号！
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