Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

Combining like terms is one of the most important skills in algebra. It allows us to simplify expressions, solve equations, and understand the structure of mathematics. This guide will take you step by step through the definition, rules, and applications of combining like terms, with plenty of examples designed for IGCSE revision.

合并同类项是代数中最重要的技能之一。它帮助我们化简表达式、解方程，并理解数学的结构。本指南将一步一步带你学习合并同类项的定义、规则和应用，并附有大量为 IGCSE 复习设计的例子。

1. What Are “Like Terms”? | 什么是“同类项”？

In algebra, a term is a number, a variable, or a product of numbers and variables. For example, 3x, −5y², 7, and 2ab are all terms. Two terms are called like terms if they have exactly the same variable parts, including the same exponents. The coefficient can be different, but the variable part must match exactly.

在代数中，项是一个数、一个变量，或数与变量的乘积。例如，3x、−5y²、7 和 2ab 都是项。如果两个项具有完全相同的变量部分，包括相同的指数，那么它们就称为同类项。系数可以不同，但变量部分必须完全一致。

For example, 4x and −7x are like terms because both have the variable x. However, 4x and 4x² are not like terms, because the exponents of x are different. Similarly, 2ab and 3ba are like terms, since multiplication is commutative: ab = ba.

例如，4x 和 −7x 是同类项，因为它们都具有变量 x。然而，4x 和 4x² 不是同类项，因为 x 的指数不同。类似地，2ab 和 3ba 是同类项，因为乘法满足交换律：ab = ba。

2. Why Do We Combine Like Terms? | 为什么我们要合并同类项？

Combining like terms simplifies an expression and makes it easier to read, evaluate, and use in solving equations. For example, the expression 5x + 3x is not in its simplest form. When we combine it, we get 8x. This is cleaner and more efficient.

合并同类项可以化简表达式，使其更易于阅读、求值和在解方程中使用。例如，表达式 5x + 3x 并不是最简形式。合并后我们得到 8x。这样更简洁、更高效。

Without combining like terms, algebraic expressions can become long and confusing. In equation solving, combining like terms on each side of the equation is often the first step before using inverse operations. It reduces the number of terms and reveals the structure of the problem.

如果不合并同类项，代数表达式会变得冗长且令人困惑。在解方程时，合并方程两边的同类项通常是使用逆运算之前的第一步。它减少了项的数量，并揭示了问题的结构。

3. Identifying Like Terms | 识别同类项

To identify like terms, look at the variable part of each term. The variable part includes both the letters and their exponents. For a term like 6x²y, the variable part is x²y. Any term with exactly x²y is a like term, regardless of its coefficient.

要识别同类项，请观察每一项的变量部分。变量部分包括字母及其指数。对于像 6x²y 这样的项，变量部分是 x²y。任何具有完全相同的 x²y 的项都是同类项，无论其系数如何。

Expression	Like terms	Not like terms
7x + 2x − 3y	7x and 2x	−3y (different variable)
5a² + a² − 3a	5a² and a²	−3a (exponent differs)
4mn + 2nm − m²n	4mn and 2nm	−m²n (different exponent)

Notice that the order of multiplication does not matter. The term 2nm is the same as 2mn. However, n² is not the same as n, and xy is not the same as x²y. Always check the exponents carefully.

注意乘法的顺序无关紧要。项 2nm 与 2mn 相同。然而，n² 与 n 不同，xy 与 x²y 也不同。请仔细检查指数。

4. Combining Addition and Subtraction | 合并加法与减法中的同类项

When combining like terms, add or subtract their coefficients. The variable part remains unchanged. For example, 3x + 5x = (3 + 5)x = 8x. Similarly, 9y − 4y = (9 − 4)y = 5y.

合并同类项时，对系数进行加法或减法运算。变量部分保持不变。例如，3x + 5x = (3 + 5)x = 8x。类似地，9y − 4y = (9 − 4)y = 5y。

When an expression contains both positive and negative terms, treat the signs as part of each term. For example: 7x − 2x + 3x = (7 − 2 + 3)x = 8x. It is often helpful to rearrange the terms so that like terms are next to each other, but remember to keep the sign in front of each term.

当表达式中同时包含正项和负项时，将符号视为每个项的一部分。例如：7x − 2x + 3x = (7 − 2 + 3)x = 8x。将同类项重新排列在相邻位置通常很有帮助，但请记住保留每项前面的符号。

4a + 6b − a + 2b = (4a − a) + (6b + 2b) = 3a + 8b

Always write the simplified expression with terms in a clear order, usually alphabetically or by descending power of the variable. This makes your work easier to check.

始终以清晰的顺序书写简化后的表达式，通常按字母顺序或按变量的降幂排列。这样更容易检查你的作答。

5. The Distributive Property and Brackets | 分配律与括号

Before combining like terms, you may need to remove brackets using the distributive property: a(b + c) = ab + ac. This is a crucial step because brackets often hide like terms that cannot be combined until the brackets are expanded.

在合并同类项之前，你可能需要使用分配律去掉括号：a(b + c) = ab + ac。这是关键的一步，因为括号往往隐藏了同类项，只有先去括号才能进行合并。

For example, simplify 3(2x + 4) + 5x. First expand: 6x + 12 + 5x. Then combine like terms: (6x + 5x) + 12 = 11x + 12.

例如，化简 3(2x + 4) + 5x。先展开：6x + 12 + 5x。然后合并同类项：(6x + 5x) + 12 = 11x + 12。

Be careful when subtracting a bracket. For example: 5x − 2(3x − 1). Expand the bracket carefully: 5x − 6x + 2 = −x + 2. The minus sign belongs to the entire bracket, so every term inside the bracket changes sign.

减去一个括号时要特别小心。例如：5x − 2(3x − 1)。仔细展开括号：5x − 6x + 2 = −x + 2。负号属于整个括号，因此括号内的每一项都要改变符号。

6. Combining Like Terms with Multiple Variables | 含多个变量的合并同类项

When an expression contains several different variables, group terms by their exact variable part. For example, simplify 2x + 3y − x + 4y. Group the x terms and the y terms separately: (2x − x) + (3y + 4y) = x + 7y.

当表达式包含几个不同的变量时，按精确的变量部分对项进行分组。例如，化简 2x + 3y − x + 4y。分别分组 x 项和 y 项：(2x − x) + (3y + 4y) = x + 7y。

Variables such as xy and x²y are distinct. For instance, 5xy + 2x²y − 3xy = 2xy + 2x²y. Notice that 5xy and −3xy combine because they both contain xy, while 2x²y remains separate because it contains x²y.

像 xy 和 x²y 这样的变量是不同的。例如，5xy + 2x²y − 3xy = 2xy + 2x²y。注意 5xy 和 −3xy 可以合并，因为它们都包含 xy，而 2x²y 保持独立，因为它包含 x²y。

It is useful to underline or circle like terms with the same marking when working with complex expressions. This prevents mistakes and helps you stay organised during exams.

在处理复杂表达式时，用相同的标记给同类项加下划线或画圈是很有用的。这可以防止错误，并帮助你在考试中保持条理。

7. Combining Fractions and Decimals | 合并分数与小数中的同类项

Like terms can have coefficients that are fractions or decimals. The same rules apply: only the coefficients are added or subtracted. For example, ½x + ⅓x = (½ + ⅓)x = 5⁄6x. You may need to find a common denominator.

同类项的系数可以是分数或小数。同样的规则适用：只对系数进行加减。例如，½x + ⅓x = (½ + ⅓)x = 5⁄6x。你可能需要找到公分母。

For decimals, 0.4x + 1.25x = 1.65x. When negative fractions appear, use the rules of signed numbers. For example, ¾y − ⅝y = (¾ − ⅝)y = (6⁄8 − 5⁄8)y = ⅛y.

对于小数，0.4x + 1.25x = 1.65x。当出现负分数时，使用符号数运算法则。例如，¾y − ⅝y = (¾ − ⅝)y = (6⁄8 − 5⁄8)y = ⅛y。

½a + 0.3a + ¼a = 0.5a + 0.3a + 0.25a = 1.05a

Always write your final answer in the form requested by the question. If the original expression uses fractions, it is often acceptable to leave the answer in fractions, but make sure you show your working clearly.

始终按照题目要求的形式书写最终答案。如果原表达式使用分数，通常可以用分数作答，但务必清楚展示你的计算过程。

8. Perimeter and Word Problems | 周长与应用题

Combining like terms is often needed when finding the perimeter of a shape with algebraic side lengths. For example, a triangle has sides x + 3, 2x − 1, and 4x + 2. The perimeter is found by adding all sides and combining like terms.

在许多情况下，求具有代数边长的图形的周长时需要合并同类项。例如，一个三角形的三边为 x + 3、2x − 1 和 4x + 2。周长通过将所有边相加并合并同类项求得。

Perimeter = (x + 3) + (2x − 1) + (4x + 2) = 7x + 4

In word problems, variables represent unknown quantities. Suppose a rectangle has length 3x and width 2x + 5. The perimeter is 2(3x) + 2(2x + 5) = 6x + 4x + 10 = 10x + 10. This simplified expression can then be used to solve for x if the perimeter is given numerically.

在应用题中，变量代表未知量。假设一个矩形的长为 3x，宽为 2x + 5。周长为 2(3x) + 2(2x + 5) = 6x + 4x + 10 = 10x + 10。如果给出了周长的具体数值，这个简化表达式就可以用来求解 x。

Word problems often involve “twice a number”, “three more than”, and “half of a number”. Translate these phrases carefully into algebra, then combine like terms to simplify the resulting expression.

应用题中常出现“一个数的两倍”、“比某数多三”和“某数的一半”等表述。将这些短语仔细翻译成代数式，然后合并同类项化简得到的表达式。

9. Common Mistakes to Avoid | 应避免的常见错误

One common mistake is combining terms that are not like terms. For example, writing 3x + 4y = 7xy is incorrect. x and y are different variables, so they cannot be combined into xy. Also, 2x + 3x² cannot become 5x³; the exponents are different.

一个常见错误是合并非同类项。例如，写 3x + 4y = 7xy 是错误的。x 和 y 是不同的变量，不能合并成 xy。同样，2x + 3x² 不能变成 5x³；指数不同。

Another mistake is forgetting the sign in front of a term when rearranging. For example, in 5x − 3x + 2, moving 2 next to 5x requires keeping the correct sign. If you move terms without their signs, you will get wrong results.

另一个错误是在重新排列时忘记项前面的符号。例如，在 5x − 3x + 2 中，将 2 移到 5x 旁边时需要保留正确的符号。如果移动项时不带符号，就会得到错误结果。

A third mistake is incorrectly applying the distributive property: −(a + b) is −a − b, not −a + b. Practice with negative brackets until you are confident. Always check whether your simplified expression makes sense by substituting a simple value.

第三个错误是错误地应用分配律：−(a + b) 等于 −a − b，而不是 −a + b。请多加练习负号括号，直到你充满信心。始终通过代入一个简单的数值来检查化简后的表达式是否合理。

10. Worked Examples for IGCSE Practice | IGCSE 练习例题

Let us go through a few typical IGCSE-style questions step by step.

让我们逐步完成几道典型的 IGCSE 风格例题。

Example 1: Simplify 6x − 2y + 3x + 5y.

例题 1：化简 6x − 2y + 3x + 5y。

Group like terms: (6x + 3x) + (−2y + 5y) = 9x + 3y.

分组同类项：(6x + 3x) + (−2y + 5y) = 9x + 3y。

Example 2: Simplify 4(2x + 1) − 3(x − 2).

例题 2：化简 4(2x + 1) − 3(x − 2)。

First expand: 8x + 4 − 3x + 6. Then combine like terms: (8x − 3x) + (4 + 6) = 5x + 10.

先展开：8x + 4 − 3x + 6。然后合并同类项：(8x − 3x) + (4 + 6) = 5x + 10。

Example 3: Simplify 5a²b − 3ab² + 2a²b + 4ab².

例题 3：化简 5a²b − 3ab² + 2a²b + 4ab²。

Group the a²b terms and the ab² terms separately: (5a²b + 2a²b) + (−3ab² + 4ab²) = 7a²b + ab².

分别分组 a²b 项和 ab² 项：(5a²b + 2a²b) + (−3ab² + 4ab²) = 7a²b + ab²。

These examples show that the key is always to identify the variable part exactly and then add or subtract the coefficients. With practice, this process becomes quick and automatic.

这些例子表明，关键在于精确识别变量部分，然后对系数进行加法或减法运算。通过练习，这个过程会变得快速而自然。

11. Summary and Key Rules | 总结与关键规则

Combining like terms is a fundamental algebraic technique. The three most important rules are: first, only combine terms with identical variable parts; second, keep the sign with its term when rearranging; third, use the distributive property to remove brackets before combining.

合并同类项是一项基础的代数技巧。最重要的三条规则是：第一，只合并变量部分完全相同的项；第二，重新排列时保留每项前面的符号；第三，先使用分配律去掉括号，再进行合并。

Expression	Simplified	Rule used
7x + 2x	9x	Add coefficients
5x − 8x	−3x	Subtract coefficients
2(x + 3) + x	3x + 6	Distribute then combine

Remember that like terms do not need to have the same coefficient. They only need the same variable part. Being able to simplify quickly will help you in all areas of IGCSE mathematics, from solving equations to working with quadratic expressions.

记住，同类项不需要具有相同的系数。它们只需要相同的变量部分。能够快速化简将帮助你在 IGCSE 数学的所有领域中表现出色，从解方程到处理二次表达式。

Finally, always check your final answer by substituting a simple number, such as 1, into both the original and the simplified expression. If the results are equal, you have almost certainly combined the terms correctly.

最后，始终通过代入一个简单的数值（如 1）来检查你的最终答案，分别代入原表达式和简化后的表达式。如果结果相等，那么你几乎肯定正确合并了项。

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Combining Like Terms | 合并同类项