Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain several terms that can be simplified. Collecting like terms is a core skill for IGCSE Mathematics because it turns a long, messy expression into a shorter, clearer one. This makes the expression easier to use in equations, graphs, and real-life problems.

在代数中，表达式往往包含若干个可以化简的项。合并同类项是 IGCSE 数学的核心技能，因为它能把冗长、杂乱的表达式改写为更短、更清晰的形式，从而更容易用于方程、图像和实际问题中。

1. What is a Term? | 什么是项？

A term in algebra is a single number, a variable, or a product of a number and one or more variables. For example, in the expression 5x + 3y – 7, the terms are 5x, 3y and -7.

代数中的“项”是一个单独的数、一个变量，或一个数与一个或多个变量的乘积。例如，在表达式 5x + 3y – 7 中，项分别是 5x、3y 和 -7。

The number part of a term is called the coefficient. In 5x, the coefficient is 5. In -3xy², the coefficient is -3 and the variable part is xy².

项中的数字部分称为系数。在 5x 中，系数是 5；在 -3xy² 中，系数是 -3，变量部分是 xy²。

Constant terms are terms with no variables, such as 4 or -9.
常数项是不含变量的项，例如 4 或 -9。

2. Identifying Like Terms | 识别同类项

Like terms are terms that have exactly the same variables raised to exactly the same powers. The coefficients do not need to be the same. For example, 3x and -5x are like terms because both contain x to the power 1.

同类项是指变量完全相同、且每个变量的指数也完全相同的项。系数不需要相同。例如，3x 和 -5x 是同类项，因为它们都含有 x 的一次幂。

Like terms (同类项)	Unlike terms (非同类项)
4x and -2x	4x and 4x²
3a²b and 7a²b	3a²b and 3ab²
-5 and 9	-5 and 5x

Notice that x and x² are not like terms because they have different powers. Even though they use the same letter, their algebraic behaviour is different.

注意，x 和 x² 不是同类项，因为它们的指数不同。虽然它们使用同一个字母，但它们的代数性质是不同的。

3. The Distributive Law | 分配律

Before collecting like terms, you may need to remove brackets. The distributive law states: a(b + c) = ab + ac. For example, 3(2x + 1) = 6x + 3.

在合并同类项之前，有时需要先去括号。分配律指出：a(b + c) = ab + ac。例如，3(2x + 1) = 6x + 3。

The same rule works with subtraction: a(b – c) = ab – ac. So 4(3x – 2) = 12x – 8.

同样的规则适用于减法：a(b – c) = ab – ac。因此 4(3x – 2) = 12x – 8。

2(x + 5) + 3(x – 1) = 2x + 10 + 3x – 3 = 5x + 7

In this example, the brackets were expanded first, then like terms were collected.

在这个例子中，先展开括号，再合并同类项。

4. Adding and Subtracting Like Terms | 同类项的加减

To collect like terms, add or subtract the coefficients and keep the variable part exactly the same. For example, 4x + 3x = 7x because 4 + 3 = 7 and the x remains unchanged.

合并同类项时，将系数相加或相减，变量部分保持不变。例如，4x + 3x = 7x，因为 4 + 3 = 7，而 x 保持不变。

5y – 2y = 3y
5y – 2y = 3y

6a² + a² = 7a²
6a² + a² = 7a²

-8xy + 3xy = -5xy
-8xy + 3xy = -5xy

Always check that the variable part is identical before adding or subtracting the coefficients.

在加减系数之前，务必检查变量部分是否完全相同。

5. Dealing with Coefficients and Signs | 系数与符号

Every term carries its sign with it. The sign belongs to the coefficient. For example, -3x + 5x = 2x because -3 + 5 = 2.

每一项的符号都属于它的系数。例如，-3x + 5x = 2x，因为 -3 + 5 = 2。

When subtracting an expression with brackets, you must change the signs of every term inside the brackets. For instance, 2x – (x – 3) = 2x – x + 3 = x + 3.

当减去一个带括号的表达式时，括号内每一项的符号都要改变。例如，2x – (x – 3) = 2x – x + 3 = x + 3。

4a – (2a – 5) = 4a – 2a + 5 = 2a + 5

This is one of the most common places where students lose marks, so practise it carefully.

这是学生最常丢分的地方之一，所以要仔细练习。

6. Simplifying Expressions with Multiple Variables | 多变量表达式化简

When an expression contains more than one variable, collect terms for each variable separately. For example, 3a + 2b – a + 4b can be rearranged as 3a – a + 2b + 4b = 2a + 6b.

当表达式包含多个变量时，需要分别合并每个变量的同类项。例如，3a + 2b – a + 4b 可改写为 3a – a + 2b + 4b = 2a + 6b。

Mixed terms must follow the same rule. In the expression 2xy + 3x² + 4xy – x², the xy terms combine and the x² terms combine: 6xy + 2x².

混合项遵循同样的规则。在表达式 2xy + 3x² + 4xy – x² 中，合并 xy 项，再合并 x² 项：6xy + 2x²。

7pq – 2p + 5pq + p = 12pq – p

Ordering the final answer is not always necessary, but it often helps to write the terms in a consistent order.

最终结果不一定要按固定顺序书写，但按一致顺序排列通常会更有帮助。

7. Combining Unlike Terms? | 不同类项能合并吗？

Unlike terms cannot be combined. The expression 2x + 3y stays as 2x + 3y because x and y are different variables. Similarly, 5x + 2x² cannot be written as a single term.

不同类项不能合并。表达式 2x + 3y 保持为 2x + 3y，因为 x 和 y 是不同变量。同样，5x + 2x² 不能写成一个单项。

Trying to combine unlike terms is the biggest algebraic error. Think of x as apples and y as oranges: two apples plus three oranges are still two apples and three oranges.

试图合并不同类项是最大的代数错误。可以把 x 想成苹果，y 想成橙子：两个苹果加三个橙子仍然是两个苹果和三个橙子。

However, unlike terms can still appear together in a simplified expression. The simplified expression is the shortest possible sum of its unlike parts.

不过，不同类项仍然可以同时出现在化简单项式中。化简后的表达式就是其不同部分所能达到的最短“和”。

8. Practical Examples and Word Problems | 实际应用与文字题

Algebraic simplification appears in perimeter problems. Suppose a rectangle has length (2x + 3) and width (x – 1). The perimeter is 2[(2x + 3) + (x – 1)] = 2(3x + 2) = 6x + 4.

代数化简常出现在周长问题中。假设一个矩形的长为 (2x + 3)，宽为 (x – 1)。它的周长为 2[(2x + 3) + (x – 1)] = 2(3x + 2) = 6x + 4。

Another example: three identical packets each contain x pencils and 2 pens. Another packet contains x pencils and 5 pens. Total items: 3(x + 2) + (x + 5) = 3x + 6 + x + 5 = 4x + 11.

另一个例子：三个相同的袋子各装有 x 支铅笔和 2 支钢笔；另一个袋子装有 x 支铅笔和 5 支钢笔。物品总数为 3(x + 2) + (x + 5) = 3x + 6 + x + 5 = 4x + 11。

These results are useful because they let us calculate the total for any value of x without rewriting the expression.

这些结果很有用，因为对于任意 x 值，都可以直接计算总数，而不需要重新写表达式。

9. Common Mistakes | 常见错误

Many students make predictable mistakes when collecting like terms. Here are the most important ones to avoid.

很多学生在合并同类项时都会犯可预见的错误。以下是需要避免的几个最重要错误。

Combining 2x and 3y incorrectly as 5xy. They are different variables, so 2x + 3y cannot be simplified.
错误地把 2x 和 3y 合并成 5xy。它们是不同变量，因此 2x + 3y 不能化简。

Confusing x and x². They are not like terms and cannot be added together.
混淆 x 与 x²。它们不是同类项，不能相加。

Dropping the negative sign when subtracting a bracket.
减去括号时丢掉了负号。

Writing 3x + 4 = 7x instead of 3x + 4. The constant 4 has no x, so it remains separate.
把 3x + 4 错写为 7x。常数 4 不含 x，所以应保持独立。

10. Exam Tips and Practice | 考试技巧与练习

In an exam, always check whether brackets need to be expanded first. Then scan the expression to group like terms before writing your final answer. It is wise to mark or underline each family of like terms.

在考试中，先检查是否需要展开括号。然后浏览表达式，将同类项分组，再写出最终答案。明智的做法是把每一组同类项标记或划线。

Try this example: simplify 5x + 3 – 2x + 7 – x² + 4x².

请尝试这个例子：化简 5x + 3 – 2x + 7 – x² + 4x²。

5x – 2x = 3x, 3 + 7 = 10, -x² + 4x² = 3x², so the answer is 3x + 10 + 3x².

Write the final expression in a clear order, often powers first, then linear terms, then constants. Practising this skill with both positive and negative coefficients will prepare you well for IGCSE questions.

最终表达式应按清晰顺序书写，通常是先幂次项，再线性项，最后常数项。通过正负系数混合练习这一技能，你就能为 IGCSE 题目做好充分准备。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Collecting Like Terms | 合并同类项