📚 Combining Like Terms | 合并同类项
When you study algebra, one of the first and most fundamental skills is simplifying expressions by combining like terms. This process makes long, messy expressions shorter and easier to work with. In IGCSE Mathematics, you will meet it again and again, from linear equations to quadratic expressions and beyond.
在学习代数时，最基础且最重要的技能之一就是通过合并同类项来化简表达式。这个过程能让冗长、杂乱的表达式变得更简洁、更容易处理。在IGCSE数学中，你会反复遇到这个技能，从线性方程到二次表达式，乃至更后面的内容。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable parts. The variable part includes both the letter(s) and their powers. For example, 3x and 7x are like terms because both contain the variable x. However, 3x and 3x² 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。变量部分包括字母及其指数。例如，3x和7x是同类项，因为它们都包含变量x。但是3x和3x²不是同类项，因为x的指数不同。
Numerical coefficients do not matter. 2a, −5a and 0.5a are all like terms. Also, constant terms (numbers without variables) are like terms: 4, −7 and 10 are all like terms.
数字系数不影响判断。2a、−5a和0.5a都是同类项。同样，常数项（不带变量的数字）也是同类项：4、−7和10都是同类项。
2. Why Combine Like Terms? | 为什么要合并同类项？
Combining like terms simplifies an expression and reduces the number of terms. This makes it easier to substitute values, solve equations, and compare expressions. For example, x + 2x + 3x can be combined into 6x, which is much simpler to evaluate.
合并同类项可以化简表达式，减少项数。这使得代入数值、解方程和比较表达式都更加容易。例如，x + 2x + 3x 可以合并成6x，计算起来简单得多。
In exam questions, leaving an unsimplified expression may lose marks. Therefore, always look for like terms and combine them before giving your final answer.
在考试中，如果答案没有化简，可能会被扣分。因此，一定要找出同类项并合并，然后才写出最终答案。
3. Identifying Like Terms | 如何识别同类项
To determine whether two terms are like, compare their variable parts. Ignore the coefficients. Check two things: the letters used, and the exponent of each letter. For instance, 4xy and −3xy are like terms, because both have xy with exponent 1 on each variable.
要判断两个项是否为同类项，比较它们的变量部分。忽略系数。检查两点：所用的字母，以及每个字母的指数。例如，4xy和−3xy是同类项，因为两者都有xy，且每个变量的指数都是1。
Be careful with the order of variables. Because multiplication is commutative, xy and yx are the same, so they are like terms. However, x²y and xy² are not like terms, because the exponents are attached to different variables.
注意变量的顺序。由于乘法满足交换律，xy和yx相同，所以它们是同类项。但是x²y和xy²不是同类项，因为指数附着在不同的变量上。
| Expression | Like terms? |
| 5a and 2a | Yes |
| 5a and 5b | No |
| 3p² and 3p | No |
| 4mn and −2nm | Yes |
4. The Distributive Property and Combining | 乘法分配律与合并
Combining like terms is based on the distributive property: ax + bx = (a + b)x. This is because x is a common factor. For example, 3x + 5x = (3 + 5)x = 8x. The same idea applies when variables have powers, but only if the powers match.
合并同类项基于乘法分配律：ax + bx = (a + b)x。这是因为x是公因式。例如，3x + 5x = (3 + 5)x = 8x。同样的原理也适用于带指数的变量，但前提是指数相同。
Let’s look at another example: 7x² − 2x² = (7 − 2)x² = 5x². Here the variable part x² is shared, so we only subtract the coefficients.
再看一个例子：7x² − 2x² = (7 − 2)x² = 5x²。这里变量部分x²是相同的，所以我们只对系数做减法。
5. Combining Positive and Negative Coefficients | 合并正系数和负系数
When an expression contains both positive and negative terms, treat the sign in front of a term as part of its coefficient. For example, in 3x − 4x + 2x, the coefficients are +3, −4 and +2. Adding them gives +1, so the result is x.
当表达式中既有正项又有负项时，把每一项前面的符号看作系数的一部分。例如，在3x − 4x + 2x中，系数分别为+3、−4和+2。将它们相加得+1，所以结果是x。
Sometimes you need to combine more than two sets of terms. Example: 2a + 3b − a + 5b. Combine the a terms: 2a − a = a. Combine the b terms: 3b + 5b = 8b. The simplified expression is a + 8b.
有时需要合并两组以上的项。例：2a + 3b − a + 5b。合并a项：2a − a = a。合并b项：3b + 5b = 8b。化简后的表达式为a + 8b。
2a + 3b − a + 5b = a + 8b
6. Terms with Different Variables Do Not Combine | 不同变量的项不能合并
A common mistake is to add terms that have different variables, such as 2x + 3y = 5xy. This is incorrect. You cannot combine x terms with y terms because they represent different quantities. Similarly, 4x + 5 cannot be simplified to 9x; the x term and the constant term are not like terms.
一个常见错误是把不同变量的项相加，例如2x + 3y = 5xy。这是错误的。不能把x项与y项合并，因为它们表示不同的数量。同样，4x + 5不能化简为9x；x项和常数项不是同类项。
However, you can leave the expression as 2x + 3y, or rewrite it as 3y + 2x if you wish. The order does not matter, but each different variable part must remain separate.
但你可以把表达式写成2x + 3y，如果愿意也可以写成3y + 2x。顺序不重要，但每个不同的变量部分必须保持独立。
7. Combining Like Terms in Geometry | 几何中的合并同类项
Simplifying expressions is very useful in geometry, especially when finding perimeters. For example, the perimeter of a rectangle with length 3x + 2 and width x + 1 is: P = 2(3x + 2) + 2(x + 1). Expand: 6x + 4 + 2x + 2. Now combine like terms: x terms give 8x, constants give 6. So P = 8x + 6.
化简表达式在几何中非常有用，尤其是在求周长时。例如，一个长为3x + 2、宽为x + 1的矩形，其周长为：P = 2(3x + 2) + 2(x + 1)。展开：6x + 4 + 2x + 2。然后合并同类项：x项得8x，常数项得6。所以P = 8x + 6。
For areas, you may need to expand brackets first. If a rectangle has sides x + 3 and x + 5, its area is (x + 3)(x + 5) = x² + 5x + 3x + 15. Combine the middle terms: x² + 8x + 15. This process will be used many times in algebra.
对于面积，你可能需要先展开括号。如果一个矩形的边长为x + 3和x + 5，其面积为(x + 3)(x + 5) = x² + 5x + 3x + 15。合并中间项：x² + 8x + 15。这个过程在代数中会多次用到。
8. Common Mistakes to Avoid | 需要避免的常见错误
Here are some errors students often make when combining like terms. First, combining unlike terms, like writing 2x + 3y = 5xy. Second, forgetting that a term like x has a coefficient of 1, so x + x = 2x, not x². Third, incorrectly combining powers: x + x² cannot be combined, and x³ + x³ = 2x³, not x⁶.
下面是学生在合并同类项时常犯的错误。第一，合并不同类的项，如把2x + 3y写成5xy。第二，忘记像x这样的项系数是1，所以x + x = 2x，而不是x²。第三，错误地合并指数：x + x²不能合并，x³ + x³ = 2x³，而不是x⁶。
Another mistake is mishandling signs. For example, 5 − 3x + 2x should first be grouped as 5 + (−3x + 2x), giving 5 − x. Students sometimes forget the subtraction sign belongs to the 3x, leading to 5 + x, which is wrong.
另一个错误是符号处理不当。例如，5 − 3x + 2x 应先分组为 5 + (−3x + 2x)，得到5 − x。学生有时忘记减号属于3x，导致得到5 + x，这是错误的。
9. Step-by-Step Method | 逐步合并的方法
To combine like terms systematically, follow these steps. Step 1: Identify and underline or circle each group of like terms. Step 2: For each group, add or subtract the coefficients, keeping the variable part unchanged. Step 3: Write the simplified expression, usually with terms in descending order of powers, and constants last.
为了系统性地合并同类项，请遵循以下步骤。第一步：找出并标记每组同类项。第二步：对每组同类项，将系数相加或相减，变量部分保持不变。第三步：写出化简后的表达式，通常按指数从高到低排列，常数项放在最后。
Let’s work through an example: 4x² − 3x + 2 + x² + 5x − 1. Group the x² terms: 4x² + x² = 5x². Group the x terms: −3x + 5x = 2x. Group the constants: 2 − 1 = 1. The answer is 5x² + 2x + 1.
让我们看一个例子：4x² − 3x + 2 + x² + 5x − 1。合并x²项：4x² + x² = 5x²。合并x项：−3x + 5x = 2x。合并常数项：2 − 1 = 1。答案为5x² + 2x + 1。
10. Practice Questions | 练习题
Try these questions yourself. 1) Simplify: 7x + 2x − 5x. 2) Simplify: 3a + 4b − a + b. 3) Simplify: 6y² − 2y² + y. 4) Simplify: 2(x + 1) + 3(x − 4). 5) Simplify: x² − x + 2x² + 3x − 1.
请自己尝试以下题目。1）化简：7x + 2x − 5x。2）化简：3a + 4b − a + b。3）化简：6y² − 2y² + y。4）化简：2(x + 1) + 3(x − 4)。5）化简：x² − x + 2x² + 3x − 1。
Answers: 1) 4x. 2) 2a + 5b. 3) 4y² + y. 4) 5x − 10. 5) 3x² + 2x − 1. If you got all of these right, you are ready for exam questions.
答案：1）4x。2）2a + 5b。3）4y² + y。4）5x − 10。5）3x² + 2x − 1。如果你全部答对了，你已经准备好应对考试题目了。
11. Exam Tips for IGCSE | IGCSE考试小贴士
In IGCSE exams, always show your working when simplifying expressions. This helps you earn method marks even if your final answer has a small error. Check your simplification by substituting a simple value, such as x = 2, into the original and simplified expressions to see if they give the same result.
在IGCSE考试中，化简表达式时一定要写出过程。即使最终答案有小错误，也能帮助你获得方法分。通过代入一个简单的数值（如x = 2）来检查化简前后的表达式是否给出相同结果，可以验证答案。
Remember that combining like terms is not only for algebraic expressions. It also appears in solving equations, rearranging formulas, and working with sequences. Mastering this skill will save you time and reduce errors across the whole paper.
记住，合并同类项不仅用于代数表达式。在解方程、重新排列公式以及处理数列时也会出现。掌握这项技能可以节省时间，并在整份试卷中减少错误。
12. Summary | 总结
To combine like terms, look for terms with the same variables and the same powers. Add or subtract their coefficients, keep the variable part unchanged, and repeat for each group of like terms. Never combine terms with different variables or different powers.
要合并同类项，你需要找变量和指数完全相同的项。将这些项的系数相加或相减，变量部分保持不变，并对每组同类项重复此过程。永远不要合并不同变量或不同指数的项。
This simple but powerful algebraic tool is essential for all higher-level mathematics. Practice until it becomes automatic, and you will find algebra much more enjoyable.
这个简单但强大的代数工具是所有更高阶数学的基础。练习到自动化的程度，你会发现代数变得更加有趣。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导