📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, simplifying an expression often begins with collecting like terms. This skill helps you turn a long or messy expression into a shorter, clearer form, which is essential for solving equations, rearranging formulas, and checking answers.
在IGCSE代数中，化简表达式通常从合并同类项开始。这项技能可以帮助你把较长或较乱的表达式变成更短、更清晰的形式，这对于解方程、变换公式和检查答案都十分重要。
1. What Are Like Terms? | 什么是同类项？
A term is a single number, variable, or product of numbers and variables. Like terms are terms that have exactly the same variable part, including the same variable letters and the same powers. Only the coefficients, the number parts, may be different.
项是指一个单独的数、变量或数与变量的乘积。同类项是指变量部分完全相同的项，包括相同的变量字母和相同的次数。只有系数，也就是数字部分，可以不同。
For example, 4x and -7x are like terms because both have the variable part x. The constant terms 3 and -8 are also like terms because neither has a variable part.
例如，4x 和 -7x 是同类项，因为它们的变量部分都是 x。常数项 3 和 -8 也是同类项，因为它们都没有变量部分。
4x and -7x are like terms; 4x and 4x² are not like terms.
2. Why Collecting Terms Matters | 为什么要合并同类项？
Collecting like terms is a form of simplification. It reduces the number of separate terms in an expression, making it easier to read and less likely to cause mistakes in later work.
合并同类项是一种化简方式。它可以减少表达式中独立项的数量，使表达式更容易阅读，也能减少后续计算中的错误。
hich can then be solved in one step. Without collecting, the equation is harder to handle.
在方程 3x + 5x = 16 中，合并同类项得到 8x = 16，这样就可以一步求解。如果不合并，方程就更难处理。
3x + 5x = 8x
3. Identifying Like Terms in Expressions | 在表达式中识别同类项
To identify like terms, ignore the coefficient and look only at the variable part. If two terms have the same letter or letters raised to the same powers, they can be collected.
识别同类项时，可以先忽略系数，只看变量部分。如果两个项的字母相同，并且字母的次数也相同，它们就可以合并。
Consider 7xy and 2xy. Both contain xy, so they are like terms. However, 7xy and 2x are not like terms because one has y but the other does not.
以 7xy 和 2xy 为例。它们都含有 xy，所以是同类项。但 7xy 和 2x 不是同类项，因为一个含 y，另一个不含 y。
| Term | Variable part | Like with 5x? |
|---|---|---|
| 5x | x | Yes |
| -3x | x | Yes |
| 5x² | x² | No |
| 5xy | xy | No |
4. Adding and Subtracting Like Terms | 同类项的加减法
To collect like terms, add or subtract their coefficients and keep the variable part unchanged. This works because the variable part is a common factor.
合并同类项时，将它们的系数相加或相减，变量部分保持不变。这是因为变量部分是公因式。
s two groups of a, giving seven groups of a. Subtraction works the same way: 9m - 4m = 5m.
例如，5a + 2a 表示五个 a 加上两个 a，得到七个 a。减法也一样：9m - 4m = 5m。
5a + 2a = 7a
9m - 4m = 5m
5. Terms with Different Powers | 不同次幂的项不能合并
Terms with the same letter but different powers are not like terms. For instance, x and x² represent different quantities, so they cannot be combined into a single term.
字母相同但次数不同的项不是同类项。例如，x 和 x² 表示不同的量，所以不能合并成一个项。
the x² terms and the x terms separately: 3x² - 2x² = x² and 4x + 5x = 9x, giving x² + 9x.
考试中常见的表达式是 3x² + 4x - 2x² + 5x。应分别合并 x² 项和 x 项：3x² - 2x² = x²，4x + 5x = 9x，得到 x² + 9x。
3x² + 4x - 2x² + 5x = x² + 9x
Remember that x + x = 2x, not x². Adding terms never changes the exponent.
请记住，x + x = 2x，而不是 x²。项相加不会改变指数。
6. Handling Negative Coefficients | 处理负系数
Negative coefficients are collected in the same way as positive ones, but you must be careful with signs. Think of the coefficient as a signed number and add the signed numbers together.
负系数的合并方法与正系数相同，但必须注意符号。可以把系数看作带符号的数，然后将这些带符号的数相加。
2, which gives -3, so the result is -3k. Likewise, -4y - 3y means -4 - 3 = -7, giving -7y.
例如，-5k + 2k 表示 -5 + 2，得到 -3，所以结果是 -3k。同样，-4y - 3y 表示 -4 - 3 = -7，得到 -7y。
-5k + 2k = -3k
-4y - 3y = -7y
7. Collecting Terms with More Than One Variable | 含多个变量的同类项合并
When a term contains more than one variable, all variables and their powers must match exactly for the terms to be like terms. The coefficient can be different, but the variable combination cannot change.
当一个项含有多个变量时，所有变量及其次数必须完全一致，这些项才是同类项。系数可以不同，但变量组合不能改变。
ot. Because multiplication is commutative, 3xy and 5yx are also like terms, since xy = yx.
例如，3xy 和 5xy 是同类项，而 3xy 和 3x 不是。由于乘法满足交换律，3xy 和 5yx 也是同类项，因为 xy = yx。
ting the xy terms: 4xy - 2xy = 2xy. The other terms remain unchanged, giving 2xy + 3x + y.
化简 4xy + 3x - 2xy + y 时，合并 xy 项：4xy - 2xy = 2xy。其他项保持不变，得到 2xy + 3x + y。
8. Removing Brackets Before Collecting | 去括号后再合并
Expressions often contain brackets that must be expanded before like terms can be collected. Use the distributive law to multiply each term inside the bracket by the factor outside.
表达式常常含有括号，必须先展开括号，然后才能合并同类项。使用分配律，将括号外的因数乘以括号内的每一项。
For example, 2(x + 3) + 3(x - 2) expands to 2x + 6 + 3x - 6. Collecting the x terms gives 5x, and the constants cancel out, so the result is 5x.
例如，2(x + 3) + 3(x - 2) 展开为 2x + 6 + 3x - 6。合并 x 项得到 5x，常数项相互抵消，所以结果是 5x。
2(x + 3) + 3(x - 2) = 5x
When a negative sign appears before a bracket, multiply every term inside by -1. Thus, -(a + b) becomes -a - b.
当括号前出现负号时，要用 -1 乘以括号内的每一项。因此，-(a + b) 变为 -a - b。
9. Common Mistakes to Avoid | 常见错误与避免方法
e terms. Writing 3x + 2y = 5xy is incorrect, because x and y are different variable parts.
一个常见错误是把不同类项当作同类项相加。写成 3x + 2y = 5xy 是错误的，因为 x 和 y 的变量部分不同。
adding. For example, 2x + 3x = 5x, not 5x². The variable part must stay exactly the same.
另一个错误是在相加时改变指数。例如，2x + 3x = 5x，而不是 5x²。变量部分必须保持完全不变。
s -a + b, not -a - b. Finally, do not forget constant terms: 3x + 2 + 4x = 7x + 2, not 7x.
学生在减去括号时还容易丢失负号。一定要把 -(a - b) 展开为 -a + b，而不是 -a - b。最后，不要忘记常数项：3x + 2 + 4x = 7x + 2，而不是 7x。
10. Exam-Style Worked Examples | 考试风格例题
Worked Example 1: Simplify 6a - 2b + 3a + 5b.
例题1：化简 6a - 2b + 3a + 5b。
s: 6a + 3a = 9a. Collect the b terms: -2b + 5b = 3b. The simplified expression is 9a + 3b.
合并 a 项：6a + 3a = 9a。合并 b 项：-2b + 5b = 3b。化简后的表达式为 9a + 3b。
6a - 2b + 3a + 5b = 9a + 3b
Worked Example 2: Simplify 4x² - 3x + 2x² + x - 5.
例题2：化简 4x² - 3x + 2x² + x - 5。
terms: 4x² + 2x² = 6x². Collect the x terms: -3x + x = -2x. The constant term remains -5.
合并 x² 项：4x² + 2x² = 6x²。合并 x 项：-3x + x = -2x。常数项保持 -5。
4x² - 3x + 2x² + x - 5 = 6x² - 2x - 5
Worked Example 3: Simplify 5(m - 2n) + 3(2m + n).
例题3：化简 5(m - 2n) + 3(2m + n)。
: 5m - 10n + 6m + 3n. Then collect m terms and n terms: 5m + 6m = 11m and -10n + 3n = -7n.
先展开两个括号：5m - 10n + 6m + 3n。然后合并 m 项和 n 项：5m + 6m = 11m，-10n + 3n = -7n。
5(m - 2n) + 3(2m + n) = 11m - 7n
11. Practice Quick Check | 快速练习
Simplify each expression by collecting like terms.
通过合并同类项化简下列每个表达式。
1. 7p + 2q - 3p + 6q
2. 5x² + 3x - 2x² + 4x
3. 2(3y - 1) + 3(y + 2)
4. 4xy - 2x + xy + 5x
Answers: 1. 4p + 8q 2. 3x² + 7x 3. 9y + 4 4. 5xy + 3x
答案：1. 4p + 8q；2. 3x² + 7x；3. 9y + 4；4. 5xy + 3x。
12. Summary and Key Tips | 总结与关键提示
Like terms have exactly the same variable part, including powers. To collect them, add or subtract only the coefficients and keep the variable part unchanged.
同类项具有完全相同的变量部分，包括次数。合并时只加减系数，变量部分保持不变。
Always expand brackets before collecting like terms, and pay special attention to negative signs. Never combine x with x², and never change an exponent when adding terms.
合并同类项前一定要展开括号，并特别注意负号。不要把 x 和 x² 合并，也不要在项相加时改变指数。
Practice by scanning the expression for matching variable groups, then rewrite the expression with one term for each group. This method helps you work quickly and accurately in IGCSE algebra questions.
练习时先扫描表达式中的相同变量组，然后把每一组项合并成一个项。这种方法能帮助你在 IGCSE 代数题中做得又快又准确。
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