Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Welcome to this TutorHao revision guide on collecting like terms — one of the most fundamental skills in GCSE and IGCSE Algebra. Mastering this topic will make solving equations, factorising, and simplifying expressions much easier.

欢迎阅读 TutorHao 复习指南：合并同类项。这是 GCSE 和 IGCSE 代数中最基础的技能之一。掌握好这项技能，会让你在解方程、因式分解和化简表达式时更加得心应手。

1. What Are ‘Terms’? | 什么是“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.

“项”指的是一个单独的数字、一个变量，或者数字与变量的乘积。例如，在表达式 3x + 5y − 2 中，项分别是 3x、5y 和 −2。

Terms can be positive or negative.

项可以是正数也可以是负数。
A term without a variable is called a constant.

没有变量的项称为常数项。
The number in front of a variable is called its coefficient.

变量前面的数字称为它的系数。

2. What Are ‘Like Terms’? | 什么是“同类项”？

Like terms are terms that have exactly the same variable part (the same letter raised to the same power). For instance, 4x and −7x are like terms because both have the variable x to the power 1.

同类项是指变量部分完全相同的项（字母相同且指数相同）。例如，4x 和 −7x 是同类项，因为它们的变量都是 x 的一次方。

Like: 3x and 5x | Unlike: 3x and 3x²

Similarly, 2y² and −8y² are like terms, but 2y² and 2y are NOT like terms because the powers of y differ.

同样的，2y² 和 −8y² 是同类项，但 2y² 和 2y 不是同类项，因为 y 的幂不同。

3. Why Do We Collect Like Terms? | 为什么要合并同类项？

Collecting like terms simplifies an expression, making it shorter and easier to work with. Simplification helps us see the structure of an expression clearly and is essential before solving equations.

合并同类项可以化简表达式，使其更简短、更易处理。化简帮助我们看清表达式的结构，并且在解方程之前是必不可少的一步。

For example, 5x + 3x can be simplified to 8x. Instead of two x-terms, we now have just one.

例如，5x + 3x 可以化简为 8x。我们不再有两个 x 项，而是只有一个。

4. The Basic Rule: Add Coefficients, Keep the Variable | 基本规则：系数相加减，变量不变

When collecting like terms, add (or subtract) the coefficients and keep the variable part unchanged.

合并同类项时，只需对系数进行加法（或减法）运算，变量部分保持不变。

6x + 2x = (6 + 2)x = 8x

9y − 4y = (9 − 4)y = 5y

Notice that the letter ‘x’ or ‘y’ remains exactly the same — we only combine the numbers in front.

注意，字母 ‘x’ 或 ‘y’ 保持不变——我们只合并前面的数字。

5. Working with Different Variable Types | 处理不同类型的变量

If an expression contains different variables, we must handle each group separately.

如果表达式中含有不同的变量，我们必须分别处理每一组。

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

We group the x-terms together and the y-terms together. Similarly, constant terms are grouped with constants.

我们将含 x 的项放在一起，含 y 的项放在一起。同样，常数项也要与常数项合并。

7a + 3 − 2a + 6 = (7a − 2a) + (3 + 6) = 5a + 9

6. Handling Like Terms with Powers (x², y³, etc.) | 处理带幂的同类项（x²、y³ 等）

Terms are like terms only if the variable and its power are identical. So x² and x² are like terms, but x² and x are not.

只有当变量及其幂完全相同时，项才是同类项。因此 x² 和 x² 是同类项，但 x² 和 x 不是。

5x² + 3x² = 8x²

4x² + 3x ≠ 7x³ (cannot be simplified)

Be careful! 4x² and 3x are not like terms because the powers of x are different. They cannot be combined.

请务必小心！4x² 和 3x 不是同类项，因为 x 的幂不同。它们不能合并。

7. Expressions with Brackets | 含括号的表达式

Sometimes we need to expand brackets before collecting like terms. Use the distributive law to remove parentheses first.

有时我们需要先去括号再合并同类项。使用分配律先去括号。

3(x + 2) + 2(x + 1) = 3x + 6 + 2x + 2

Now collect like terms:

现在合并同类项：

= (3x + 2x) + (6 + 2) = 5x + 8

Expanding first is a crucial step — collecting like terms should only take place after all brackets have been removed.

先去括号是关键步骤——只有在所有括号都展开之后才进行合并同类项。

8. Subtracting Terms: Be Careful with Negative Signs | 减法项：小心负号

When subtracting terms, pay close attention to negative signs. Subtracting a positive term moves left on the number line; subtracting a negative term moves right.

做减法时，要特别注意负号。减去一个正项在数轴上向左移动；减去一个负项则向右移动。

7x − 2x = 5x

4a − (−3a) = 4a + 3a = 7a

Subtracting a negative is equivalent to adding the positive. This is a common trap in exams.

减去一个负数等于加上它的正值。这是考试中常见的陷阱。

9. Collecting Like Terms in Geometry Problems | 在几何题中合并同类项

Collecting like terms often appears in geometry, especially when finding perimeters of shapes with unknown side lengths.

合并同类项经常出现在几何问题中，特别是在求含未知边长的图形周长时。

Example: A rectangle has sides (3x + 2) and (x + 4). Find its perimeter.

例：一个长方形的边长分别为 (3x + 2) 和 (x + 4)，求它的周长。

Perimeter = 2(3x + 2) + 2(x + 4)

= 6x + 4 + 2x + 8

= 8x + 12

The final simplified expression gives us the perimeter in terms of x.

最终化简后的表达式给出了用 x 表示的周长。

10. Simplifying with Fractions and Decimals | 分数和小数的化简

Like terms can also involve fractional or decimal coefficients. The same rule applies — combine the coefficients, keep the variable.

同类项也可以包含分数或小数系数。同样规则适用——合并系数，保留变量。

½x + ¾x = (½ + ¾)x = 1¼x

0.5y + 0.25y = 0.75y

⅓x − ⅙x = (⅓ − ⅙)x = ⅙x

Converting fractions to a common denominator is helpful when the fractions are different.

当分数不同时，将它们转化为公分母会很有帮助。

11. Common Mistakes to Avoid | 需要避免的常见错误

Let us examine some common errors students make when collecting like terms.

让我们来看一些学生在合并同类项时常犯的错误。

Mistake: 3x + 4y = 7xy. Wrong! Different variables cannot be combined.

错误：3x + 4y = 7xy。错误！不同的变量不能合并。
Mistake: 2x + 3x² = 5x³. Wrong! Different powers cannot be combined.

错误：2x + 3x² = 5x³。错误！不同幂不能合并。
Mistake: 5 − 3x = 2x. Wrong! A constant and a variable term are not like terms.

错误：5 − 3x = 2x。错误！常数项和变量项不是同类项。
Mistake: x + x = x². Wrong! x + x = 2x, since we add coefficients.

错误：x + x = x²。错误！x + x = 2x，因为我们要加系数。

12. Practice Problems | 练习

Try these problems on your own before checking the answers.

先自己尝试这些题目，再核对答案。

1) 7x + 3x − 2x	2) 5a + 2b − a + 3b
3) 4x² + 3x + 2x² + x	4) 2(x + 3) + 4(x − 1)
5) 6y − 3y + 2y − y	6) ½x + ¼x + 2x

Answers: 1) 8x, 2) 4a + 5b, 3) 6x² + 4x, 4) 6x + 2, 5) 4y, 6) 2¾x

答案：1) 8x，2) 4a + 5b，3) 6x² + 4x，4) 6x + 2，5) 4y，6) 2¾x

If you obtained all six answers correctly, you are ready for more advanced algebra. If not, review the sections where you struggled and try again!

如果六道题全部正确，说明你已经准备好学习更高级的代数。如果有错误，请复习相应的小节，再次练习！

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Collecting Like Terms | 合并同类项