📚 Collecting Like Terms | 合并同类项
Collecting like terms is one of the most important skills in algebra. It allows us to simplify expressions by combining terms that have the same variable parts, making equations easier to solve and understand.
合并同类项是代数中最重要的技能之一。它让我们通过合并具有相同变量部分的项来化简表达式，使方程更容易求解和理解。
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. In an expression, terms are separated by + or − signs. For example, in 3x + 2y − x + 5, there are four terms: 3x, 2y, −x, and 5.
项是一个数字、一个变量，或数字与变量的乘积。在一个表达式中，项由 + 或 − 符号分隔。例如，在 3x + 2y − x + 5 中，有四个项：3x、2y、−x 和 5。
Like terms are terms that have exactly the same variable part. The variables and their exponents must match. Only the numerical coefficients may be different. For instance, 4x and 7x are like terms, but 4x and 4x² are not like terms because the powers of x are different.
同类项是指变量部分完全相同的项。变量及其指数必须相同，只有数字系数可以不同。例如，4x 和 7x 是同类项，但 4x 和 4x² 不是同类项，因为 x 的幂不同。
Like terms: 5a and −2a; 3xy and 8xy; 7 and −4
Not like terms: 5a and 5b; 3xy and 3x²y; 7 and 4x
2. Why Do We Collect Like Terms? | 为什么要合并同类项
Collecting like terms simplifies an expression. A simplified expression has fewer terms and is easier to evaluate, rearrange, and solve. It also helps you see the overall structure of an algebraic statement clearly.
合并同类项可以化简表达式。化简后的表达式项数更少，更容易求值、变形和求解。它还能帮助你清楚地看到代数式的整体结构。
is not just a shortcut; it is based on the distributive property: 5x + 3x = (5 + 3)x = 8x.
考虑 5x + 3x。我们可以不写两个独立的项，而是将它们合并为 8x。这不仅仅是一种捷径，而是基于乘法分配律：5x + 3x = (5 + 3)x = 8x。
4a + 2b − a + 3b = (4a − a) + (2b + 3b) = 3a + 5b
3. Step-by-Step Method | 分步方法
To collect like terms successfully, follow a clear routine. First, identify all the terms in the expression. Second, group the like terms together, either mentally or by underlining them in matching patterns. Third, add or subtract the coefficients of each group. Finally, write the simplified expression with each variable group appearing once.
要成功合并同类项，遵循一个清晰的步骤。首先，识别表达式中的所有项。其次，将同类项分组，可以在心中分组，也可以用相同的下划线标记它们。第三，对每一组进行系数的加减。最后，写出化简后的表达式，每个变量组只出现一次。
terms: 2x + 4x = 6x. Group the y terms: 3y − y = 2y. The simplified expression is 6x + 2y.
例如，化简 2x + 3y + 4x − y。将 x 项分组：2x + 4x = 6x。将 y 项分组：3y − y = 2y。化简后的表达式是 6x + 2y。
Original: 7c + 2 + 5c − 3 → Group: 7c + 5c and 2 − 3 → Simplified: 12c − 1
4. Working with Addition and Subtraction | 加法与减法中的合并
² + 5x² = 8x², not 8x⁴. The exponent of x must not be changed when you collect like terms.
在加减同类项时，只有系数发生变化，变量部分保持完全不变。例如，3x² + 5x² = 8x²，而不是 8x⁴。合并同类项时，x 的指数不能改变。
before each term as belonging to that term. For example, 6x − 2x + 3x = (6 − 2 + 3)x = 7x.
减法也是如此。9p − 4p = 5p。如果同时有加法和减法，把每个项前面的符号视为该项的一部分。例如，6x − 2x + 3x = (6 − 2 + 3)x = 7x。
4k + 7k − 2k = (4 + 7 − 2)k = 9k
5. Dealing with Different Variables | 处理不同变量
An expression can contain several different variables, such as x, y, and z. Terms with different variables are not like terms and cannot be combined. For example, 2x + 3y cannot be simplified to 5xy. The variable parts are different, so the expression has already been simplified as far as possible.
一个表达式可以包含多个不同的变量，例如 x、y 和 z。具有不同变量的项不是同类项，不能合并。例如，2x + 3y 不能化简为 5xy。由于变量部分不同，这个表达式已经尽量化简了。
ables does not matter for multiplication, but the set of variables must be identical. Thus 3ab and 4ba are like terms.
然而，像 2xy 和 5xy 这样的项是同类项，因为两者都包含完全相同的乘积 x × y。变量相乘的顺序并不重要，但变量集合必须完全相同。因此，3ab 和 4ba 是同类项。
m + 2n + 3m + 4p = 4m + 2n + 4p
6. Combining Terms with Coefficients and Constants | 合并带系数项与常数项
nts are like terms with each other. You can combine −4 + 7 = 3 because both are constants.
常数是不含变量的固定数字，例如 5、−2 或 0.5。所有常数彼此之间都是同类项。你可以合并 −4 + 7 = 3，因为两者都是常数。
ame rule applies: keep the variable exactly the same and add or subtract the coefficients.
系数可以是整数、分数、小数或负数。例如，½x + ¼x = ¾x，0.6y + 0.3y = 0.9y。规则相同：保持变量完全不变，对系数进行加减。
2x + 5 + 3x − 1 = (2x + 3x) + (5 − 1) = 5x + 4
7. Negative Signs and Subtracting Expressions | 负号与表达式相减
When subtracting an entire expression, you must distribute the negative sign to every term inside the brackets. For example, (5x + 3) − (2x + 1) becomes 5x + 3 − 2x − 1, which simplifies to 3x + 2. If you forget to change the sign of every term, your answer will be wrong.
减去整个表达式时，必须将负号分配给括号内的每一个项。例如，(5x + 3) − (2x + 1) 变为 5x + 3 − 2x − 1，化简为 3x + 2。如果忘记改变其中某一项的符号，答案就会出错。
= 2x + 3. Notice that −3 becomes +3 because subtracting a negative is the same as adding.
要特别注意像 4x − (2x − 3) 这样的表达式。括号外的负号会改变括号内两个项的符号：4x − 2x + 3 = 2x + 3。注意 −3 变成 +3，因为减去一个负数等于加上它。
(7a − 2) − (3a + 4) = 7a − 2 − 3a − 4 = 4a − 6
8. Perimeter and Word Problems | 周长与应用题
Collecting like terms appears frequently in geometry. For example, the perimeter of a triangle is the sum of its side lengths. If the sides are x + 3, 2x + 5, and 3x − 2, then the perimeter is (x + 3) + (2x + 5) + (3x − 2). Collecting like terms gives 6x + 6.
合并同类项经常出现在几何中。例如，三角形的周长是三条边长之和。如果三边分别是 x + 3、2x + 5 和 3x − 2，那么周长是 (x + 3) + (2x + 5) + (3x − 2)。合并同类项得到 6x + 6。
plain pencils and 2 coloured pencils, the total cost is 4p + 2(3p) = 4p + 6p = 10p cents.
应用题通常要求你先建立表达式，然后化简。假设商店出售两种铅笔：普通铅笔每支 p 分，彩色铅笔每支是普通铅笔的 3 倍。如果一个学生买了 4 支普通铅笔和 2 支彩色铅笔，总费用是 4p + 2(3p) = 4p + 6p = 10p 分。
Rectangle: length = 2x + 1, width = x − 2 → Perimeter = 2(2x + 1) + 2(x − 2) = 6x − 2
9. Common Mistakes to Avoid | 常见错误避免
One common mistake is combining terms with different exponents. 2x + 3x² cannot be simplified because one term is linear and the other is quadratic. Another mistake is changing the variable when adding coefficients: 2x + 3x is never 5x²; it is 5x.
常见错误之一是合并指数不同的项。2x + 3x² 不能化简，因为一个是线性项，另一个是二次项。另一个错误是在加法中改变变量：2x + 3x 绝不是 5x²，而是 5x。
mbined: 4x − 2x + 7 + 3 = 2x + 10. Leaving the constants separate is not fully simplified.
学生也常常忘记合并常数项。在表达式 4x + 7 − 2x + 3 中，7 和 3 都是常数，所以应该合并：4x − 2x + 7 + 3 = 2x + 10。让常数项分开不是完全化简。
When written in a general form, a simplified expression usually lists variable terms first, then constants. For example, write 2x + 10 rather than 10 + 2x, although both are mathematically correct.
在书写一般形式时，化简后的表达式通常先写变量项，再写常数项。例如，写 2x + 10 而不是 10 + 2x，尽管两者在数学上都是正确的。
10. Practice Questions | 练习
Try the following questions before checking the answers. These questions cover all the key ideas from this article.
在查看答案之前，请先尝试以下问题。这些问题涵盖本文的所有关键要点。
| Question | Answer |
| 1. Simplify 5x + 3x | 8x |
| 2. Simplify 2a + 5b − a + 3b | a + 8b |
| 3. Simplify 4x² + 2x² − 3x | 6x² − 3x |
| 4. Simplify (3y + 2) + (2y − 5) | 5y − 3 |
| 5. Simplify (4x + 1) − (x − 2) | 3x + 3 |
| 6. Simplify ½m + ¼m + 2 | ¾m + 2 |
If you made an error on question 5, review the section on negative signs. Remember to subtract every term inside the brackets, not just the first one.
如果你在第 5 题出错，请复习关于负号的部分。记住要减去括号内的每一个项，而不只是第一个项。
11. Summary | 总结
Collecting like terms means combining terms that have the same variable part. Constants are also like terms with each other. When you combine like terms, add or subtract only the coefficients and keep the variable part unchanged. Different variables and different powers cannot be combined.
合并同类项是指将具有相同变量部分的项合并在一起。常数之间也可视为同类项。合并同类项时，只对系数进行加减，并保持变量部分不变。不同的变量和不同的幂不能合并。
Always work carefully with negative signs, especially when subtracting a whole expression. With regular practice, collecting like terms becomes a fast and reliable tool for solving algebra problems at IGCSE level.
处理负号时务必小心，尤其是在减去整个表达式时。通过经常练习，合并同类项将成为你解决 IGCSE 代数问题的快速而可靠的工具。
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