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 IGCSE Mathematics. It appears everywhere, from simple algebraic expressions to complex equation solving. This article explains the concept clearly, provides step-by-step examples, and shows you how to avoid common mistakes.

合并同类项是IGCSE数学中最重要的一项技能。它出现在从简单代数式到复杂方程求解的各个部分。本文将清晰地解释这一概念，提供逐步示例，并帮助你避免常见错误。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable parts, including the same variables raised to the same powers. The coefficients, or the numbers in front, do not need to be the same.

同类项是指变量部分完全相同的项，即相同变量且指数相同。系数，也就是数字部分，不必相同。

For example, 3x and 5x are like terms because both contain the variable x to the power 1. Similarly, 2xy and -7xy are like terms because both contain the product xy with no exponents. However, 4x and 4x² are not like terms, because the powers of x are different.

例如，3x与5x是同类项，因为两者都含有变量x且指数为1。类似地，2xy与-7xy是同类项，因为两者都含有乘积xy而没有指数。但是，4x与4x²不是同类项，因为x的幂不同。

Like Terms 同类项	Unlike Terms 非同类项
7a and 3a	7a and 7a²
4xy and -xy	4xy and 4x
2p² and 9p²	2p² and 2p

2. Why Combine Like Terms? | 为什么需要合并同类项

Combining like terms simplifies an expression so that it is shorter and easier to understand. For instance, the expression 5x + 3x – 2x can be written as a single term 6x.

合并同类项能化简代数式，使其更简短、更易于理解。例如，5x + 3x – 2x可以化简为单项式6x。

This skill is essential when solving equations, simplifying formulas, and working with geometry problems. A simpler expression also reduces the chance of making arithmetic errors in later steps.

这一技能在解方程、化简公式以及处理几何问题时都必不可少。化简后的表达式还能降低后续步骤中算术失误的概率。

In IGCSE examinations, questions often begin with an instruction such as “Simplify” or “Collect like terms”. Mastering this step ensures you gain full marks before moving on to more difficult skills.

在IGCSE考试中，题目常常以“化简”或“合并同类项”为指令。掌握这一步能确保你在进入更高难度技能之前先拿到满分。

3. The Fundamental Rule | 合并的基本原则

To combine like terms, add or subtract the coefficients and keep the variable part unchanged. The variable part acts like a label that tells you which terms can be grouped together.

合并同类项时，将系数相加或相减，并保持变量部分不变。变量部分就像一个标签，告诉你哪些项可以归为一组。

ax + bx = (a + b)x

For example, 4x + 6x = (4 + 6)x = 10x. The letter x remains exactly the same; only the coefficients are added.

例如，4x + 6x = (4 + 6)x = 10x。字母x完全保持不变，只有系数相加。

The same rule works for subtraction: ax – bx = (a – b)x. So 9y – 2y = 7y.

这条规则同样适用于减法：ax – bx = (a – b)x。因此9y – 2y = 7y。

4. Positive and Negative Terms | 正项与负项的合并

Expressions often contain a mixture of positive and negative terms. To combine them correctly, imagine walking along a number line or think of the signs as belonging to each term.

代数式中常常同时含有正项和负项。要正确合并，可以想象在数轴上行走，或者把符号看作属于每一项的一部分。

Consider the expression 7x – 3x + 2x. The coefficients are +7, -3, and +2. Adding them gives 7 – 3 + 2 = 6, so the simplified expression is 6x.

考虑表达式7x – 3x + 2x。系数分别为+7、-3和+2。将它们相加得到7 – 3 + 2 = 6，因此化简结果为6x。

7x – 3x + 2x = (7 – 3 + 2)x = 6x

Notice that the negative sign belongs to the term 3x. You should never leave a sign “floating”; always attach it to the coefficient that follows.

注意负号属于3x这一项。绝不能把符号“悬空”；始终要将符号与后面的系数绑定在一起。

5. Worked Example: Single Variable | 示例：单变量合并

Let us simplify a longer expression step by step: 4x + 5x – 3x + 2x.

我们来逐步化简一个较长的表达式：4x + 5x – 3x + 2x。

Step 1: Identify all terms that contain x. They are 4x, 5x, -3x, and 2x.

第一步：找出所有含x的项。它们是4x、5x、-3x和2x。

Step 2: Add the coefficients: 4 + 5 – 3 + 2 = 8.

第二步：将系数相加：4 + 5 – 3 + 2 = 8。

Step 3: Attach the variable x to the result.

第三步：将变量x写在结果后面。

4x + 5x – 3x + 2x = 8x

Check your answer by substituting a value, say x = 2. The original expression gives 4(2) + 5(2) – 3(2) + 2(2) = 8 + 10 – 6 + 4 = 16, and 8(2) = 16. The simplification is correct.

通过代入一个数值来检查答案，例如x = 2。原表达式为4(2) + 5(2) – 3(2) + 2(2) = 8 + 10 – 6 + 4 = 16，而8(2) = 16。化简正确。

6. Worked Example: Multiple Variables | 示例：多变量合并

When an expression contains more than one variable, combine each variable separately. For example, simplify 3a + 4b – a + 2b.

当表达式含有多个变量时，应分别合并每个变量。例如，化简3a + 4b – a + 2b。

First group the a terms: 3a – a = 2a. Then group the b terms: 4b + 2b = 6b.

先合并a项：3a – a = 2a。再合并b项：4b + 2b = 6b。

3a + 4b – a + 2b = 2a + 6b

Never try to combine 2a and 6b, because they are unlike terms. The final answer must keep them separate.

绝不能尝试合并2a与6b，因为它们不是同类项。最终答案必须将它们分开书写。

For a more challenging example, simplify 5p² + 3p – 2p² + 7p. The p² terms combine to 5p² – 2p² = 3p², and the p terms combine to 3p + 7p = 10p. The answer is 3p² + 10p.

再看一个更具挑战性的例子：化简5p² + 3p – 2p² + 7p。p²项合并为5p² – 2p² = 3p²，p项合并为3p + 7p = 10p。答案为3p² + 10p。

5p² + 3p – 2p² + 7p = 3p² + 10p

7. Fractions and Decimals | 分数与小数系数

Like terms with fractional or decimal coefficients follow exactly the same rule. Combine the numerical coefficients using normal fraction or decimal arithmetic.

含有分数或小数系数的同类项遵循完全相同的规则。运用普通的分数或小数运算来合并数值系数。

For example, (1/2)x + (1/4)x = (2/4)x + (1/4)x = (3/4)x. The variable x stays unchanged.

例如，(1/2)x + (1/4)x = (2/4)x + (1/4)x = (3/4)x。变量x保持不变。

½x + ¼x = ¾x

Decimal coefficients work the same way: 0.3y + 0.5y = 0.8y, and -2.5z + z = -1.5z.

小数系数也同样适用：0.3y + 0.5y = 0.8y，且-2.5z + z = -1.5z。

When you see a coefficient like x on its own, remember it really means 1x. This helps when adding terms such as x + 0.5x = 1.5x.

当你看到单独的x这样的系数时，请记住它实际上等于1x。这有助于处理x + 0.5x = 1.5x这类计算。

8. Common Mistakes | 常见错误

The most frequent mistake is confusing x with x². They are not like terms and cannot be combined. Writing 3x + 4x² = 7x² is wrong.

最常见的错误是混淆x与x²。它们不是同类项，不能合并。写成3x + 4x² = 7x²是错误的。

Another common error is dropping negative signs. For the expression 5x – 3x, the correct result is 2x, not 8x or 2. You must keep the variable in the answer.

另一个常见错误是遗漏负号。对于5x – 3x，正确结果是2x，而不是8x或2。答案中必须保留变量。

Students also sometimes forget to write the coefficient 1. When the result is 1x, we write simply x. For instance, 5x – 4x = x, not 1 (leaving out the variable) and not 0 (if computed incorrectly).

学生有时还会忘记书写系数1。当结果为1x时，我们通常简写为x。例如，5x – 4x = x，而不是1（漏写变量），也不是0（如果计算错误）。

Finally, be careful with expressions like 2ab + 3ba. Because multiplication is commutative, ab and ba are the same, so they are like terms and combine to 5ab.

最后，注意像2ab + 3ba这样的表达式。因为乘法满足交换律，ab与ba相同，所以它们是同类项，合并为5ab。

9. Geometry Connection | 与几何的联系

Combining like terms appears naturally in geometry, especially when finding perimeters and areas.

合并同类项在几何中自然出现，尤其是在求周长和面积时。

Consider a triangle whose side lengths are 2x, 3x, and 4x. Its perimeter is 2x + 3x + 4x = 9x.

考虑一个三角形，三边长分别为2x、3x和4x。其周长为2x + 3x + 4x = 9x。

Perimeter = 2x + 3x + 4x = 9x

For a rectangle with length 5a and width 2a, the perimeter is 5a + 2a + 5a + 2a = 14a. The area is 5a × 2a = 10a², which also involves combining powers of the same base.

对于长为5a、宽为2a的矩形，其周长为5a + 2a + 5a + 2a = 14a。面积为5a × 2a = 10a²，这也涉及到同底数幂的合并。

These examples show that combining like terms is not just a mechanical exercise. It gives simple algebraic answers that can be used in further calculations.

这些例子表明，合并同类项不仅仅是机械练习，它能给出简洁的代数答案，供后续计算使用。

10. Exam-Style Questions | 考试题型

IGCSE questions often ask you to simplify an expression or to identify which expression is equivalent to a given one. Let us look at a typical example.

IGCSE题目通常要求你化简表达式，或判断哪个表达式与给定表达式等价。我们来看一个典型例子。

Simplify: 6x + 3y – 2x + 4y.

化简：6x + 3y – 2x + 4y。

Group the x terms: 6x – 2x = 4x. Group the y terms: 3y + 4y = 7y. The answer is 4x + 7y.

合并x项：6x – 2x = 4x。合并y项：3y + 4y = 7y。答案为4x + 7y。

6x + 3y – 2x + 4y = 4x + 7y

Another common question gives you a multiple-choice list. For example, which expression equals 5x + 2 – 3x + 8? The x terms give 2x and the constants give 2 + 8 = 10, so the answer is 2x + 10.

另一种常见题型是给出多个选项。例如，哪个表达式等于5x + 2 – 3x + 8？x项合并为2x，常数项合并为2 + 8 = 10，因此答案为2x + 10。

Remember to check whether “constant” terms, numbers without variables, are also like terms. All pure numbers are like terms and can be combined together.

请记住，“常数项”也就是没有变量的数字，同样属于同类项。所有纯数字都可以互相合并。

11. Practice Problems | 巩固练习

Try these exercises yourself before looking at the answers. They cover the main situations you will meet in the exam.

请先自己尝试以下练习，再查看答案。它们涵盖了考试中会遇到的主要情形。

Simplify 8a + 3a – 2a.

化简8a + 3a – 2a。
Simplify 5m + 2n – 3m + 4n.

化简5m + 2n – 3m + 4n。
Simplify 7x – 4x² + 2x + 3x².

化简7x – 4x² + 2x + 3x²。
Simplify (2/3)y + (1/6)y.

化简(2/3)y + (1/6)y。
Simplify 0.4p + 3q – 0.6p + q.

化简0.4p + 3q – 0.6p + q。
The sides of a triangle are x, 2x + 3, and 4x – 1. Find its perimeter in simplified form.

三角形三边分别为x、2x + 3和4x – 1。求其周长的最简形式。

Answers: 1. 9a. 2. 2m + 6n. 3. 9x – x². 4. (5/6)y. 5. -0.2p + 4q. 6. 7x + 2.

答案：1. 9a。2. 2m + 6n。3. 9x – x²。4. (5/6)y。5. -0.2p + 4q。6. 7x + 2。

12. Summary | 总结

Combining like terms means adding or subtracting the coefficients of terms that have identical variable parts. You keep the variable part exactly the same.

合并同类项就是对变量部分完全相同的项进行系数相加或相减，并保持变量部分完全不变。

Always separate different variables and different powers. Remember the order of operations and take care with negative signs and coefficients of 1.

始终对不同变量和不同次幂分别处理。注意运算顺序，小心处理负号和系数1。

With regular practice, combining like terms becomes quick and automatic. It is a foundational skill that supports algebra, equations, coordinate geometry, and many other topics in IGCSE Mathematics.

通过定期练习，合并同类项会变得快速而熟练。它是支撑IGCSE数学中代数、方程、坐标几何及许多其他专题的基础技能。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Combining Like Terms | 合并同类项