📚 Combining Like Terms | 合并同类项
In IGCSE Mathematics, combining like terms is one of the most important early algebra skills. It allows you to simplify expressions so that they are shorter, clearer, and easier to work with when solving equations, substituting values, or expanding brackets. This article explains what like terms are, how to identify them, and how to combine them accurately using positive and negative coefficients.
在 IGCSE 数学中，合并同类项是早期代数中最重要的基本技能之一。它能让代数式变得更短、更清晰，并且在解方程、代入数值或去括号时更容易处理。本文将解释什么是同类项、如何识别同类项，以及如何在有正负系数的情况下正确合并它们。
1. What Are Like Terms? | 什么是同类项？
A term is a single number, a single variable, or a product of numbers and variables. For example, 3x, -5y, 7, 2ab, and x² are all terms. Like terms are terms that have exactly the same variable part, including the same powers. The numerical part, called the coefficient, may be different.
一个项可以是一个数字、一个变量，或者数字与变量的乘积。例如 3x、-5y、7、2ab 和 x² 都是项。同类项是指变量部分完全相同，包括相同的幂。数字部分，也就是系数，可以不同。
For example, 3x and 5x are like terms because both contain the variable x raised to the power 1. Likewise, 2x² and -7x² are like terms because both contain x². However, 3x and 3x² are not like terms because the powers of x are different.
例如，3x 和 5x 是同类项，因为它们都含有变量 x 的一次方。同样，2x² 和 -7x² 也是同类项，因为它们都含有 x²。但是，3x 和 3x² 不是同类项，因为 x 的幂不同。
2. Why Combine Like Terms? | 为什么要合并同类项？
Combining like terms simplifies an expression into its shortest possible form. This makes it easier to evaluate the expression for different values, compare expressions, and solve equations. In an exam, a simplified answer is usually required for full marks.
合并同类项可以将一个代数式化简为最简形式。这使得代数式在代入不同数值时更容易求值，也更容易比较和用于解方程。在考试中，通常要求给出化简后的答案才能获得满分。
For example, instead of writing 2x + 3x + 4, you can write 5x + 4. Both expressions have the same value for any x, but the second form is simpler and contains fewer operations.
例如，与其写成 2x + 3x + 4，不如写成 5x + 4。对于任意 x，这两个代数式的值都相同，但第二种形式更简单，运算也更少。
3. Identifying Like Terms: Variables and Powers | 识别同类项：变量与幂
To decide whether two terms are like terms, ignore the coefficient and look only at the variable part. Ask two questions: Are the variables the same? Are the powers of each variable the same? If both answers are yes, the terms are like terms.
判断两个项是否为同类项时，忽略系数，只看变量部分。问两个问题：变量是否相同？每个变量的幂是否相同？如果两个答案都是“是”，那么它们就是同类项。
For instance, 4x and -11x are like terms. The expressions 6y² and 3y² are like terms. However, 5x and 5y are not like terms because the variables are different. Similarly, x and x³ are not like terms because the powers differ.
例如，4x 和 -11x 是同类项。6y² 和 3y² 是同类项。然而，5x 和 5y 不是同类项，因为变量不同。同样，x 和 x³ 也不是同类项，因为幂不同。
For terms with more than one variable, every variable and every power must match. For example, 3xy and 7xy are like terms, and 2a²b and -5a²b are like terms. But 3ab and 3a²b are not like terms because the power of a is different.
对于含有多个变量的项，每个变量和每个幂都必须匹配。例如，3xy 和 7xy 是同类项，2a²b 和 -5a²b 是同类项。但是 3ab 和 3a²b 不是同类项，因为 a 的幂不同。
4. Adding and Subtracting Like Terms | 同类项的加减
When you add or subtract like terms, keep the variable part unchanged and add or subtract the coefficients only. This works because the variable part is common to both terms.
当你对同类项进行加减时，保持变量部分不变，只对系数进行加减。这是因为变量部分是两项共有的。
3x + 5x = (3 + 5)x = 8x
3x + 5x = (3 + 5)x = 8x
s perform the arithmetic with the coefficients, then attach the variable part once at the end.
同样，9a - 4a = (9 - 4)a = 5a。始终先对系数进行运算，然后在末尾写一次变量部分即可。
e applies: 2x² + 6x² = 8x². Do not change the power when adding or subtracting like terms.
对于二次项，规则相同：2x² + 6x² = 8x²。在加减同类项时，不要改变幂。
5. Combining with Negative Coefficients | 负系数的合并
Negative coefficients are handled in exactly the same way, but you must be careful with signs. Treat each term as having a sign attached to its coefficient.
负系数的处理方式完全相同，但你必须小心符号。把每一项看作带有自己的系数符号。
7y - 3y = (7 - 3)y = 4y
7y - 3y = (7 - 3)y = 4y
+ 5p = 3p, but -4m - 6m = -10m. Use the rules for adding and subtracting directed numbers.
如果合并后的系数变为负数，化简后的项就是负数。例如，-2p + 5p = 3p，但 -4m - 6m = -10m。请使用正负数的加减规则。
n of the second term. In 3x - 9x, the second coefficient is -9, so 3x - 9x = -6x, not +6x.
一个常见错误是忽略第二项的符号。在 3x - 9x 中，第二项的系数是 -9，所以 3x - 9x = -6x，而不是 +6x。
6. Multiple Variable Terms | 多变量项
Terms with more than one variable can also be combined if all variables and all powers match exactly. Because multiplication is commutative, the order of the variables does not matter.
含有多个变量的项也可以合并，只要所有变量和所有幂都完全匹配。由于乘法满足交换律，变量的顺序并不重要。
2ab + 5ab = 7ab
2ab + 5ab = 7ab
y + 2yx = 5xy. However, xy and x²y are not like terms because the power of x is different.
你可能还会见到 xy 和 yx。它们是同类项，因为 xy = yx。因此，3xy + 2yx = 5xy。然而，xy 和 x²y 不是同类项，因为 x 的幂不同。
Do not combine 2ab with 3a or 3b, because the variable parts are not the same. The term 2ab contains both a and b, while 3a contains only a.
不要把 2ab 与 3a 或 3b 合并，因为变量部分不同。项 2ab 同时含有 a 和 b，而 3a 只含有 a。
7. Constants and Mixed Expressions | 常数与混合表达式
Constants are numbers with no variable part. Only constants can be combined with other constants. They cannot be combined with terms that contain variables.
常数是没有变量部分的数字。只有常数才能与其他常数合并。它们不能与含有变量的项合并。
n an expression such as 2x + 3 + 5x - 1, combine the x terms and the constants separately.
例如，5 + 3 = 8，4 - 9 = -5。在 2x + 3 + 5x - 1 这样的代数式中，要分别合并 x 项和常数项。
2x + 3 + 5x - 1 = (2x + 5x) + (3 - 1) = 7x + 2
2x + 3 + 5x - 1 = (2x + 5x) + (3 - 1) = 7x + 2
When simplifying mixed expressions, you may gather like terms in any order, but it is helpful to group them first. Write the variable terms together and the constants together before combining.
化简混合代数式时，可以按任意顺序合并同类项，但最好先把它们分组。先将含变量的项写在一起，常数项写在一起，然后再合并。
8. Brackets and Expanding First | 先去括号再合并
If an expression contains brackets, you must usually expand the brackets before collecting like terms. Use the distributive law to multiply each term inside the bracket by the factor outside.
如果代数式中含有括号，通常必须先展开括号，然后再合并同类项。使用分配律，将括号外的因数乘以括号内的每一项。
3(x + 2) = 3x + 6
3(x + 2) = 3x + 6
After expanding all brackets, the expression may contain like terms that can be collected. For example, 3(x + 2) + 2(x - 1) becomes 3x + 6 + 2x - 2, which simplifies to 5x + 4.
展开所有括号后，表达式可能含有可以合并的同类项。例如，3(x + 2) + 2(x - 1) 变为 3x + 6 + 2x - 2，再化简为 5x + 4。
3(x + 2) + 2(x - 1) = 3x + 6 + 2x - 2 = 5x + 4
3(x + 2) + 2(x - 1) = 3x + 6 + 2x - 2 = 5x + 4
9. Common Mistakes | 常见错误
copies of x added together, so x + x = 2x. The expression x × x is the one that equals x².
一个非常常见的错误是把 x + x 写成 x²。这是不正确的，因为 x + x 表示两个 x 相加，所以 x + x = 2x。而 x × x 才等于 x²。
Another error is trying to combine terms with different variables or powers, such as 2x + 3y or x + x². These terms cannot be added into a single term, so the expression must remain as it is or as a sum of separate terms.
另一个错误是试图合并变量或幂不同的项，例如 2x + 3y 或 x + x²。这些项不能合并为一个项，因此表达式必须保持原样，或者写成分开的项的和。
Students also sometimes forget to keep the power unchanged. When combining 2x² and 3x², the answer is 5x², not 5x⁴. The exponent does not change when adding or subtracting like terms.
学生有时还会忘记保持幂不变。当合并 2x² 和 3x² 时，答案是 5x²，而不是 5x⁴。在加减同类项时，指数不会改变。
10. Worked Examples | 例题精讲
Worked Example 1: Simplify 4a + 7b - 2a + 3b.
例题 1：化简 4a + 7b - 2a + 3b。
First, group the a terms and the b terms: (4a - 2a) + (7b + 3b). Then combine each group: 2a + 10b.
首先，将 a 项和 b 项分组：(4a - 2a) + (7b + 3b)。然后合并每一组：2a + 10b。
4a + 7b - 2a + 3b = 2a + 10b
4a + 7b - 2a + 3b = 2a + 10b
Worked Example 2: Simplify 5x² + 3x - 2x² + 7x.
例题 2：化简 5x² + 3x - 2x² + 7x。
Group like terms: (5x² - 2x²) + (3x + 7x). The result is 3x² + 10x. Note that x² terms and x terms are kept separate because their powers are different.
合并同类项：(5x² - 2x²) + (3x + 7x)。结果是 3x² + 10x。注意 x² 项和 x 项要分开，因为它们的幂不同。
5x² + 3x - 2x² + 7x = 3x² + 10x
5x² + 3x - 2x² + 7x = 3x² + 10x
11. Practice Checklist | 练习清单
Use the following checklist when simplifying any algebraic expression. First, expand brackets if necessary. Second, identify all like terms by comparing variable parts and powers. Third, group like terms together. Fourth, add or subtract the coefficients. Finally, check that no unlike terms have been combined and that all powers remain unchanged.
化简任何代数式时，使用以下清单。首先，必要时展开括号。第二，通过比较变量部分和幂来识别所有同类项。第三，将同类项分组。第四，加减系数。最后，检查没有合并不同类项，并且所有幂保持不变。
- Expand brackets using the distributive law. / 使用分配律展开括号。
- Look for terms with identical variable parts. / 寻找变量部分完全相同的项。
- Keep x, x², xy, and constant terms separate. / 将 x、x²、xy 和常数项分开。
- Check negative signs carefully. / 仔细检查负号。
- Do not change exponents when adding or subtracting. / 加减时不要改变指数。
12. Summary | 总结
Combining like terms is the process of simplifying an expression by adding or subtracting the coefficients of terms that have exactly the same variable part. The variable part, including all powers, stays unchanged.
合并同类项是通过对变量部分完全相同的项进行系数加减来化简代数式的过程。变量部分，包括所有幂，保持不变。
Remember that constants combine with constants, x terms combine with x terms, and x² terms combine only with other x² terms. Terms with different variables or different powers cannot be combined into one single term.
记住，常数与常数合并，x 项与 x 项合并，x² 项只能与其他 x² 项合并。变量不同或幂不同的项不能合并为一个项。
Mastering this skill is essential for success in IGCSE algebra. It forms the foundation for expanding brackets, factorising, solving linear equations, and working with more complex algebraic fractions.
掌握这一技能对于在 IGCSE 代数中取得成功至关重要。它是展开括号、因式分解、解一元一次方程以及处理更复杂的分式代数的基础。
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