Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Collecting like terms is one of the most fundamental skills in algebra. It allows us to simplify algebraic expressions by combining terms that share the same variable and the same power. This skill appears in nearly every IGCSE Mathematics exam paper, so mastering it is essential for success.

合并同类项是代数中最基础的技能之一。它让我们能够将具有相同变量和相同指数的项组合起来，从而简化代数表达式。这项技能几乎出现在每一份 IGCSE 数学试卷中，因此掌握它对取得好成绩至关重要。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part. This means they have the same variables raised to the same powers. For example, 3x and 5x are like terms because both contain the variable x to the first power. Similarly, 2x² and -7x² are like terms because both contain x squared.

同类项是指变量部分完全相同的项。这意味着它们具有相同的变量以及相同的指数。例如，3x 和 5x 是同类项，因为它们都包含一次方的变量 x。类似地，2x² 和 -7x² 是同类项，因为它们都包含 x 的平方。

Unlike terms have different variables or different powers. For instance, 3x and 3x² are NOT like terms because the powers of x differ. Likewise, 4x and 4y are not like terms because the variables differ.

非同类项具有不同的变量或不同的指数。例如，3x 和 3x² 不是同类项，因为 x 的指数不同。同样地，4x 和 4y 也不是同类项，因为变量不同。

Like terms: 3x and 5x | 2a²b and -8a²b | 7 and 4

Unlike terms: 3x and 3x² | 4x and 4y | 2x and 2xy

A constant term (a number without a variable) is considered a “like term” with other constants. So 7 and 4 can be combined to give 11.

常数项（没有变量的数字）被视为与其他常数是”同类项”。因此 7 和 4 可以合并得到 11。

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

When collecting like terms, we add or subtract the coefficients (the numbers in front of the variables) while keeping the variable part exactly the same. For example, 3x + 5x = 8x. We simply add 3 and 5 to get 8, and the x remains unchanged.

合并同类项时，我们对系数（变量前面的数字）进行加法或减法运算，而变量部分保持不变。例如，3x + 5x = 8x。我们只需将 3 和 5 相加得到 8，x 保持不变。

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

This rule applies to all powers and all variables. If the variable part is identical, we can combine the terms regardless of how many terms there are.

该规则适用于所有指数和所有变量。只要变量部分相同，无论有多少项，我们都可以将它们合并。

3. Combining Terms with Subtraction | 含有减法的合并

When some terms have negative coefficients, we must be careful with signs. For example, 9x – 4x = 5x. Here we subtract 4 from 9 to get 5. The key is to treat the sign as part of the coefficient.

当某些项具有负系数时，我们必须注意符号。例如，9x – 4x = 5x。这里我们用 9 减去 4 得到 5。关键在于将符号视为系数的一部分。

9x – 4x = (9 – 4)x = 5x

For expressions with three or more terms, such as 6x – 2x + 10x, we compute: 6 – 2 + 10 = 14, so the answer is 14x. Always combine all like terms in one step to avoid missing any.

对于包含三个或更多项的表达式，如 6x – 2x + 10x，我们计算：6 – 2 + 10 = 14，因此答案是 14x。始终一步合并所有同类项，以免遗漏任何一项。

4. Simplifying Expressions with Multiple Variable Types | 含多种变量的化简

Many algebraic expressions contain different variables. For example, 5x + 3y – 2x + 7y. Here, x terms and y terms are separate groups. We combine 5x – 2x = 3x, and 3y + 7y = 10y. The result is 3x + 10y.

许多代数表达式包含不同的变量。例如，5x + 3y – 2x + 7y。其中，x 项和 y 项是独立的两组。我们合并 5x – 2x = 3x，以及 3y + 7y = 10y。结果是 3x + 10y。

5x + 3y – 2x + 7y = 3x + 10y

Remember: 3x and 3y are not like terms, so they cannot be combined. The final expression 3x + 10y is already in its simplest form.

请记住：3x 和 3y 不是同类项，因此不能合并。最终表达式 3x + 10y 已经是最简形式。

5. Combining Terms with Different Powers | 含不同指数项的合并

Terms with the same variable but different powers are NOT like terms. For example, x² and x are different. We cannot combine 4x² + 3x into a single term. However, 4x² + 3x² = 7x².

相同变量但指数不同的项不是同类项。例如，x² 和 x 是不同的。我们不能将 4x² + 3x 合并为一项。然而，4x² + 3x² = 7x²。

4x² + 3x² = 7x²

4x² + 3x cannot be simplified further

When simplifying expressions like 3x² + 5x – 2x² + 7x, we group the x² terms and the x terms separately: (3x² – 2x²) + (5x + 7x) = x² + 12x.

当化简 3x² + 5x – 2x² + 7x 这类表达式时，我们分别组合 x² 项和 x 项：(3x² – 2x²) + (5x + 7x) = x² + 12x。

6. Dealing with Constants | 处理常数项

Constants are numbers without variables, and they can be combined with each other. For example, 4x + 7 + 2x – 3. Combine the x terms: 4x + 2x = 6x. Combine the constants: 7 – 3 = 4. The result is 6x + 4.

常数项是没有变量的数字，它们可以相互合并。例如，4x + 7 + 2x – 3。合并 x 项：4x + 2x = 6x。合并常数项：7 – 3 = 4。结果是 6x + 4。

4x + 7 + 2x – 3 = 6x + 4

Always check whether the constant terms have been fully combined. A common mistake is to forget to combine all constants, especially when there are several of them.

始终检查常数项是否已完全合并。一个常见错误是忘记合并所有常数，尤其是当常数项有多个时。

7. Terms with Coefficients of 1 or -1 | 系数为 1 或 -1 的项

When a variable appears without a written coefficient, the coefficient is understood to be 1. For example, x means 1x, and -x means -1x. This is important when combining: x + 3x = 4x, and -x + 5x = 4x.

当变量前没有写出系数时，系数默认为 1。例如，x 表示 1x，而 -x 表示 -1x。这在合并时很重要：x + 3x = 4x，且 -x + 5x = 4x。

x + 3x = 4x

-x + 5x = 4x

When simplifying x – 2x, the coefficient of x is 1, so 1 – 2 = -1, giving -x. Do not forget to write the negative sign in the final answer.

化简 x – 2x 时，x 的系数为 1，因此 1 – 2 = -1，得到 -x。不要忘记在最终答案中写出负号。

8. Real-World Application | 实际应用

Collecting like terms is used in geometry, physics, and economics. For example, finding the perimeter of a rectangle with sides 2x + 3 and x – 1: Perimeter = 2(2x + 3) + 2(x – 1) = (4x + 6) + (2x – 2) = 6x + 4.

合并同类项广泛应用于几何、物理和经济学中。例如，求边长为 2x + 3 和 x – 1 的矩形周长：周长 = 2(2x + 3) + 2(x – 1) = (4x + 6) + (2x – 2) = 6x + 4。

Perimeter = 2(2x + 3) + 2(x – 1) = 6x + 4

In algebra word problems, simplifying expressions at the very beginning often makes the rest of the problem much easier to solve.

在代数应用题中，从一开始就化简表达式往往能使后续问题更容易求解。

9. Common Mistakes to Avoid | 常见错误避免

One common mistake is combining terms with different powers, such as writing x² + x = x³. This is wrong; x² and x are not like terms. Another mistake is incorrectly handling negative signs, such as computing 5x – 3x as -2x instead of 2x.

一个常见错误是合并不同指数的项，例如写成 x² + x = x³。这是错误的；x² 和 x 不是同类项。另一个错误是错误处理负号，例如把 5x – 3x 计算为 -2x 而不是 2x。

A third mistake is forgetting to include the sign of a term when moving it in the expression. For example, in 7x – 3 + 2x – 5, the constants are -3 and -5, so their combination is -8, not +8.

第三个错误是在表达式中移动项时忘记包括该项的符号。例如，在 7x – 3 + 2x – 5 中，常数项是 -3 和 -5，因此它们的合并结果是 -8，而不是 +8。

Incorrect	Correct	Explanation
x² + x = x³	Cannot simplify	Different powers
5x – 3x = -2x	5x – 3x = 2x	5 – 3 = 2
4 + 3x = 7x	Cannot simplify	4 is a constant, 3x is a variable term

10. Step-by-Step Strategy | 分步策略

To simplify any algebraic expression by collecting like terms, follow these steps:

要通过合并同类项化简任何代数表达式，请遵循以下步骤：

Identify all terms with the same variable and the same power, and group them together.

找出所有具有相同变量和相同指数的项，并将它们归为一组。
Combine each group by adding or subtracting the coefficients.

通过加减系数来合并每组。
Combine all constant terms separately.

将所有的常数项分别合并。
Write the simplified expression, usually arranging terms in descending order of power.

写出化简后的表达式，通常按指数降序排列各项。

For example, simplify 8y – 3 + 2y² – 5y + 7 – y²:

例如，化简 8y – 3 + 2y² – 5y + 7 – y²：

y² terms: 2y² – y² = y²

y terms: 8y – 5y = 3y

constants: -3 + 7 = 4

Final answer: y² + 3y + 4

11. Practice Questions | 练习题

Try these problems on your own. Simplify each of the following expressions:

请独立尝试以下问题。化简下列各表达式：

Simplify: 7a + 2a – 3a

化简：7a + 2a – 3a
Simplify: 5x + 3y – 2x + 8y

化简：5x + 3y – 2x + 8y
Simplify: 4m² + 6m – 2m² + 3m – 5

化简：4m² + 6m – 2m² + 3m – 5
Simplify: 2x + 3 + 5x – 7 + x

化简：2x + 3 + 5x – 7 + x

Answers: 1) 6a 2) 3x + 11y 3) 2m² + 9m – 5 4) 8x – 4

答案：1) 6a 2) 3x + 11y 3) 2m² + 9m – 5 4) 8x – 4

12. Summary | 总结

Collecting like terms is the process of simplifying algebraic expressions by combining terms that have the same variable and the same power. Add the coefficients and keep the variable part unchanged. Constants combine with constants, and variables with different powers or different letters must remain separate.

合并同类项是通过组合具有相同变量和相同指数的项来化简代数表达式的过程。将系数相加，保持变量部分不变。常数与常数合并，不同指数或不同字母的变量必须保持独立。

This fundamental skill will support your work in solving equations, expanding brackets, factorising, and working with algebraic fractions throughout your IGCSE course. Practice it until it becomes second nature.

这项基础技能将贯穿你的 IGCSE 课程，支持你在解方程、展开括号、因式分解和代数分数等方面的学习。多加练习，直到它成为你的自然反应。

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Collecting Like Terms | 合并同类项