Collecting Like Terms & Simplifying Expressions | 合并同类项与代数式化简

📚 Collecting Like Terms & Simplifying Expressions | 合并同类项与代数式化简

In algebra, the process of collecting like terms is one of the most fundamental skills you will ever learn. It allows us to simplify expressions, solve equations more easily, and understand the structure of mathematical formulas. This topic appears throughout the IGCSE Mathematics syllabus, whether you are working with linear expressions, quadratics, or even functions.

在代数中，合并同类项是最基础、最重要的技能之一。它帮助我们化简代数式、更轻松地解方程，并理解数学公式的结构。无论你学习的是线性表达式、二次式还是函数，这一主题始终贯穿于IGCSE数学考纲之中。

1. What Are Terms and Like Terms? | 什么是项与同类项？

A term is a single number, a variable, or a product of numbers and variables. For example, in the expression 3x + 5y − 2, the terms are 3x, 5y and −2. Terms are separated by addition and subtraction signs.

项（term）是一个单独的数字、一个变量，或数字与变量的乘积。例如，在表达式 3x + 5y − 2 中，项分别是 3x、5y 和 −2。项与项之间由加号和减号分隔。

Like terms are terms that have exactly the same variable part raised to the same power. For instance, 4x and −7x are like terms because both contain the variable x to the power of 1. Similarly, 2x² and 5x² are like terms, but 2x² and 5x are not.

同类项（like terms）是指变量部分完全相同且指数也相同的项。例如，4x 和 −7x 是同类项，因为它们都含有一次变量 x；同样，2x² 和 5x² 是同类项，但 2x² 和 5x 不是同类项。

3a and −8a → like terms (same variable a)
3a and −8a² → NOT like terms (different powers)
3a and −8b → NOT like terms (different variables)
6ab and −4ab → like terms

3a 和 −8a → 是同类项（变量相同，皆为 a）
3a 和 −8a² → 不是同类项（指数不同）
3a 和 −8b → 不是同类项（变量不同）
6ab 和 −4ab → 是同类项

2. The Basic Rule of Collecting Like Terms | 合并同类项的基本法则

To collect like terms, simply add or subtract their coefficients while keeping the variable part unchanged. The coefficient is the number in front of the variable. For example, 5x + 3x = 8x; the coefficient 5 plus the coefficient 3 gives 8, and the variable x remains the same.

合并同类项的方法非常简单：将它们的系数相加或相减，变量部分保持不变。系数是变量前面的数字。例如，5x + 3x = 8x；系数 5 加系数 3 得 8，变量 x 保持不变。

ax + bx = (a + b)x

Similarly, ax − bx = (a − b)x. This rule applies to any like terms, whether they contain positive or negative coefficients.

同理，ax − bx = (a − b)x。这个法则适用于任何同类项，无论它们的系数是正数还是负数。

Example 1: Simplify 7x + 2x − 5x.

示例 1：化简 7x + 2x − 5x。

Solution: All three terms contain x to the power of 1, so they are like terms. Combine the coefficients: 7 + 2 − 5 = 4. Therefore the simplified expression is 4x.

解答：三个项都含有一次变量 x，因此它们是同类项。合并系数：7 + 2 − 5 = 4。所以化简结果为 4x。

3. Combining Terms with Different Powers | 含不同幂次项的合并

It is crucial to distinguish between different powers of the same variable. x, x² and x³ are NOT like terms. Each power group must be collected separately.

区分同一变量的不同幂次极为重要。x、x² 和 x³ 不是同类项。每个幂次组必须分别合并。

2x + 3x² + 4x − x² = (2x + 4x) + (3x² − x²) = 6x + 2x²

Example 2: Simplify 5x² + 3x − 2x² + 7x.

示例 2：化简 5x² + 3x − 2x² + 7x。

Identify like terms: 5x² and −2x²; 3x and 7x.
Combine x² terms: 5x² − 2x² = 3x².
Combine x terms: 3x + 7x = 10x.
Answer: 3x² + 10x.

识别同类项：5x² 和 −2x²；3x 和 7x。
合并 x² 项：5x² − 2x² = 3x²。
合并 x 项：3x + 7x = 10x。
答案：3x² + 10x。

4. Expressions with Multiple Variables | 含多个变量的表达式

When an expression contains more than one variable, treat each variable family as its own group. For example, terms with x are collected together, terms with y are collected together, and terms with xy are collected together.

当表达式含有多个变量时，将每个变量家族视为一个独立的组。例如，含 x 的项合并在一起，含 y 的项合并在一起，含 xy 的项合并在一起。

3x + 4y − 2x + 5y = (3x − 2x) + (4y + 5y) = x + 9y

Example 3: Simplify 6a + 3b − 2a + b − a.

示例 3：化简 6a + 3b − 2a + b − a。

Solution: Group the a terms: 6a − 2a − a = 3a. Group the b terms: 3b + b = 4b. The simplified expression is 3a + 4b.

解答：合并 a 项：6a − 2a − a = 3a。合并 b 项：3b + b = 4b。化简结果为 3a + 4b。

Watch out for terms like xy and yx: because multiplication is commutative, xy and yx are the same, so they are like terms.

注意 xy 与 yx 这样的项：由于乘法满足交换律，xy 与 yx 相同，因此它们是同类项。

5. Simplifying with Brackets | 含括号的化简

When brackets appear in an expression, the first step is to expand them using the distributive law: a(b + c) = ab + ac. After expanding, collect like terms.

当表达式中出现括号时，第一步是使用分配律展开括号：a(b + c) = ab + ac。展开之后，再合并同类项。

a(b + c) = ab + ac

Example 4: Simplify 3(2x + 5) + 4(x − 2).

示例 4：化简 3(2x + 5) + 4(x − 2)。

Solution: First expand both brackets:

解答：首先展开两个括号：

3(2x + 5) = 6x + 15

4(x − 2) = 4x − 8

Then combine: 6x + 15 + 4x − 8 = (6x + 4x) + (15 − 8) = 10x + 7.

然后合并：6x + 15 + 4x − 8 = (6x + 4x) + (15 − 8) = 10x + 7。

Remember: when subtracting a bracket, distribute the negative sign to every term inside the bracket. For example, 5x − 2(3x − 1) = 5x − 6x + 2 = −x + 2.

请记住：减去一个括号时，必须将负号分配给括号内的每一项。例如，5x − 2(3x − 1) = 5x − 6x + 2 = −x + 2。

6. Fractional Coefficients | 分数系数

Sometimes terms have fractional coefficients. The same rule applies: add or subtract the fractions, keep the variable unchanged. For example:

有时项的系数是分数。同样的法则适用：将分数相加或相减，保持变量不变。例如：

½x + ⅓x = (½ + ⅓)x = ⁵⁄₆x

Example 5: Simplify ¼y + ¾y − ½y.

示例 5：化简 ¼y + ¾y − ½y。

Solution: Combine the coefficients: ¼ + ¾ − ½ = 1 − ½ = ½. The answer is ½y.

解答：合并系数：¼ + ¾ − ½ = 1 − ½ = ½。答案为 ½y。

When dealing with decimal coefficients, the procedure is identical. Simply add or subtract the decimals.

当系数是小数时，过程完全相同，直接将小数相加或相减即可。

7. Combining Multiplication and Division | 合并乘除运算

Some expressions involve products of variables. When simplifying terms like 2x × 3x, multiply the numbers and multiply the variables separately: 2 × 3 = 6 and x × x = x², so 2x × 3x = 6x².

有些表达式涉及变量的乘积。化简像 2x × 3x 这样的项时，分别将数字相乘、变量相乘：2 × 3 = 6，x × x = x²，因此 2x × 3x = 6x²。

Example 6: Simplify 4a × 2b + 3a × a.

示例 6：化简 4a × 2b + 3a × a。

4a × 2b = 8ab
3a × a = 3a²
These two products are NOT like terms (ab and a² have different variable parts), so the final answer is 8ab + 3a².

4a × 2b = 8ab
3a × a = 3a²
这两个乘积不是同类项（ab 与 a² 的变量部分不同），所以最终答案为 8ab + 3a²。

8. Common Mistakes and Pitfalls | 常见错误与陷阱

Students often make similar mistakes when collecting like terms. Let us look at the most common ones and how to avoid them.

学生在合并同类项时常犯类似的错误。让我们看看最常见的几种，以及如何避免。

Common Error \| 常见错误	Why It Is Wrong \| 错误原因	Correct Answer \| 正确答案
3x + 4y = 7xy	x and y are different variables; they cannot be combined.	3x + 4y
5x² − x = 4x	x² and x have different powers.	5x² − x
x + x = x²	x + x = 2x, not x².	2x
2(3x − 4) = 6x − 4	The 2 must multiply both 3x and −4.	6x − 8

常见错误	错误原因	正确答案
3x + 4y = 7xy	x 和 y 是不同的变量，不能合并。	3x + 4y
5x² − x = 4x	x² 与 x 的幂次不同。	5x² − x
x + x = x²	x + x = 2x，而不是 x²。	2x
2(3x − 4) = 6x − 4	2 必须同时乘以 3x 和 −4。	6x − 8

A good habit is to underline or circle each family of like terms in different colours before combining them. This reduces careless mistakes, especially in long expressions.

一个好习惯是在合并之前，用不同的颜色给每一组同类项做下划线或圈出标记。这样可以减少粗心错误，尤其是对于较长的表达式。

9. Exam Tips and Common Question Types | 考试技巧与常见题型

In IGCSE exams, collecting like terms often appears in the early parts of Paper 2 and Paper 4 (or Paper 1 and Paper 3 depending on your tier). Questions may ask you to simplify an expression, expand brackets and simplify, or even substitute values after simplification.

在IGCSE考试中，合并同类项常出现在试卷的第2部分（或根据级别不同，出现在第1/第3部分）的前几题。题目可能要求化简一个表达式、展开括号并化简，甚至要求在化简后进行代入数值求解。

Simplify 4(2x + 3) − 5(x − 1)

Step-by-step approach for such questions:

解答此类问题的分步方法：

Step 1: Expand any brackets using the distributive law.
Step 2: Identify like terms (same variable, same power).
Step 3: Combine the coefficients of each like-term group.
Step 4: Write the final answer in a neat, simplified form.

第一步：使用分配律展开所有括号。
第二步：识别同类项（同变量、同幂次）。
第三步：将每组同类项的系数合并。
第四步：写出整洁、化简后的最终答案。

Solution to the above question: 4(2x + 3) = 8x + 12; 5(x − 1) = 5x − 5; therefore 8x + 12 − (5x − 5) = 8x + 12 − 5x + 5 = 3x + 17.

上述题目的解答：4(2x + 3) = 8x + 12；5(x − 1) = 5x − 5；因此 8x + 12 − (5x − 5) = 8x + 12 − 5x + 5 = 3x + 17。

10. Practice Questions | 练习与挑战

Now it is your turn. Try these questions without looking at the answers below.

现在轮到你练习了。先尝试独立完成下列题目，再看下方答案。

1. 5x + 7x − 3x

2. 2a² + 3a − a² + 5a

3. 4(2x + 1) + 3(x − 2)

4. 6m − 2(3m + 4)

5. ½(4x + 6) + ⅓(9x − 3)

Answers and worked solutions:

答案与详细解答：

Question 1: 5x + 7x − 3x = (5 + 7 − 3)x = 9x.

第1题：5x + 7x − 3x = (5 + 7 − 3)x = 9x。

Question 2: Group like terms: 2a² − a² = a²; 3a + 5a = 8a. Answer: a² + 8a.

第2题：合并同类项：2a² − a² = a²；3a + 5a = 8a。答案：a² + 8a。

Question 3: Expand: 4(2x + 1) = 8x + 4; 3(x − 2) = 3x − 6. Combine: 8x + 4 + 3x − 6 = 11x − 2.

第3题：展开：4(2x + 1) = 8x + 4；3(x − 2) = 3x − 6。合并：8x + 4 + 3x − 6 = 11x − 2。

Question 4: Expand: 2(3m + 4) = 6m + 8. Then 6m − (6m + 8) = 6m − 6m − 8 = −8.

第4题：展开：2(3m + 4) = 6m + 8。然后 6m − (6m + 8) = 6m − 6m − 8 = −8。

Question 5: Expand: ½(4x + 6) = 2x + 3; ⅓(9x − 3) = 3x − 1. Combine: 2x + 3 + 3x − 1 = 5x + 2.

第5题：展开：½(4x + 6) = 2x + 3；⅓(9x − 3) = 3x − 1。合并：2x + 3 + 3x − 1 = 5x + 2。

11. Real-World Connections | 实际应用联系

Simplification is not just an abstract classroom exercise. Engineers simplify complex algebraic models before calculating stress on bridges; economists combine terms to simplify demand and supply functions; computer scientists use algebraic simplification to optimise code. The ability to recognise patterns and combine like terms quickly is a transferable skill across all STEM fields.

化简不仅仅是课堂上的抽象练习。工程师在计算桥梁受力之前需要先化简复杂的代数模型；经济学家通过合并项来简化需求与供给函数；计算机科学家利用代数化简来优化程序代码。快速识别模式并合并同类项的能力是贯穿所有STEM领域的可迁移技能。

In real exam contexts, even if a question seems long, the underlying skill is often just collecting like terms. Once you master it, many “hard” questions become much easier.

在真实考试中，即使题目看起来很长，其核心技能往往就是合并同类项。一旦掌握了它，许多“难题”就会变得简单得多。

12. Summary | 总结

To collect like terms successfully, remember the golden rules: (1) only combine terms with the exact same variable and power; (2) add or subtract coefficients, never variables; (3) always expand brackets first; (4) be careful with negative signs, especially when subtracting brackets; (5) write your final answer with terms in a consistent order.

要成功合并同类项，请记住黄金法则：（1）只合并变量和幂次完全相同的项；（2）只能加减系数，绝不可加减变量；（3）始终先展开括号；（4）小心负号，尤其是减去括号时；（5）最终答案中的项要按一致顺序书写。

Practice daily, and this skill will become second nature. With consistent effort, simplifying expressions will be one of the easiest parts of your IGCSE Mathematics exam.

每天练习，这项技能就会成为你的本能反应。只要坚持不懈，化简代数式将成为你IGCSE数学考试中最简单的一部分。

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Collecting Like Terms & Simplifying Expressions | 合并同类项与代数式化简