Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Algebra is a language of symbols, and one of the most essential skills in that language is simplification. When we combine terms that share the same variable form, we are “collecting like terms.” This article explains the rules step by step, with clear examples for IGCSE students.

代数是符号的语言，而其中最重要的技能之一就是化简。当我们把具有相同变量形式的项合并起来时，就是在做“合并同类项”。本文将为 IGCSE 学生逐步解释规则，并配以清晰的例题。

1. Understanding Algebraic Terms | 理解代数项

An algebraic term is a product of a number and one or more variables. For example, 5x, -3y², 4ab, and 7 are all algebraic terms. The numerical part is called the coefficient, and the variable part is called the ‘literal part’ or variable factor.

一个代数项是一个数与一个或多个变量的乘积。例如 5x、-3y²、4ab 和 7 都是代数项。其中数字部分叫做系数，变量部分叫做“字母部分”或变量因子。

Term = Coefficient × Variable part

项 = 系数 × 变量部分

For instance, in the term -4x³, the coefficient is -4 and the variable part is x³. The sign of the term is included in the coefficient. Understanding this structure is crucial before we can collect like terms.

例如，在项 -4x³ 中，系数是 -4，变量部分是 x³。项的符号包含在系数之中。先理解这一结构，我们才能进行合并同类项。

2. What Are ‘Like Terms’? | 什么是“同类项”？

Like terms are terms that have exactly the same variable part, including the same powers (exponents). Only the coefficients may differ. For example, 2x and 5x are like terms because both contain x to the power 1. Similarly, 3x² and -7x² are like terms because both contain x².

同类项是指变量部分完全相同的项，包括相同的幂（指数）。只有系数可以不同。例如 2x 和 5x 是同类项，因为它们都含 x 的一次幂。同样，3x² 和 -7x² 是同类项，因为它们都含 x²。

Constant terms are also considered like terms. The numbers 4, -9, and 0.5 are all like terms because they have no variable part at all.

常数项也被视为同类项。数字 4、-9 和 0.5 都是同类项，因为它们完全没有变量部分。

Like Terms	Unlike Terms
2x, 3x, -0.5x	2x, 2y
4y², -6y²	4y², 4y
7, -2, 4	7, 7x

Notice that 4y² and 4y are not like terms because the powers of y are different. This distinction is the key to successful simplification.

注意 4y² 和 4y 不是同类项，因为 y 的幂不同。这种区分是成功化简的关键。

3. The Process of Collecting Like Terms | 合并同类项的步骤

To collect like terms, first identify all terms that share the same variable part. Then add or subtract their coefficients. Keep the variable part unchanged. This process is sometimes called “simplifying an expression.”

合并同类项时，首先要找出所有变量部分相同的项，然后对这些项的系数进行加法或减法，变量部分保持不变。这个过程有时也叫做“化简表达式”。

Consider the expression: 4x + 3x. Both terms contain x, so we add the coefficients: 4 + 3 = 7. The result is 7x.

考虑表达式：4x + 3x。两项都含 x，所以我们把系数相加：4 + 3 = 7。结果是 7x。

4x + 3x = (4 + 3)x = 7x

The variable part x remains the same; we never multiply or change it when collecting like terms.

变量部分 x 保持不变；在合并同类项时，我们绝不对变量部分进行乘法或改变。

4. Adding and Subtracting Like Terms | 同类项的加法和减法

When adding and subtracting like terms, treat the sign of each term as part of its coefficient. For example, in the expression 6a – 2a, subtract 2 from 6 to get 4a.

在加减同类项时，要把每一项的符号视为其系数的一部分。例如，在表达式 6a – 2a 中，用 6 减 2 得到 4a。

6a – 2a = 4a

For expressions with more than two like terms, combine them one step at a time or use a vertical method. Example: 5b + 3b – 2b = (5 + 3 – 2)b = 6b.

对于含两个以上同类项的表达式，可以逐步合并，也可以用竖式方法。例如：5b + 3b – 2b = (5 + 3 – 2)b = 6b。

Always pay attention to negative signs. The term -4x + 7x means starting at -4 and moving up 7, giving +3. So -4x + 7x = 3x.

始终注意负号。项 -4x + 7x 表示从 -4 开始向上移动 7，得到 +3。因此 -4x + 7x = 3x。

5. Dealing with Coefficients and Variables | 处理系数和变量

Sometimes terms have a coefficient of 1 or -1, such as x or -y. Remember that 1x is written as x, and -1y is written as -y. When collecting terms, keep this in mind: x + 2x = 3x, not 2x².

有时项的系数是 1 或 -1，例如 x 或 -y。记住 1x 写作 x，-1y 写作 -y。合并时特别注意：x + 2x = 3x，而不是 2x²。

Variables can be different letters. Terms with different variable parts cannot be combined. For example, 2x + 3y cannot be simplified further because x and y are different variables.

变量可能是不同的字母。变量部分不同的项不能合并。例如，2x + 3y 不能再化简，因为 x 和 y 是不同的变量。

If a variable part has exponents, the exponents must match. Examples of like terms: 2x² and 4x²; examples of unlike terms: 2x² and 2x³. You cannot add x² and x³ together.

如果变量部分有指数，指数必须相同。同类项的例子：2x² 和 4x²；非同类项的例子：2x² 和 2x³。x² 和 x³ 不能相加。

6. Collecting Terms in Longer Expressions | 在较长表达式中合并项

Longer expressions may contain multiple groups of like terms. The best approach is to rearrange the terms so that like terms are adjacent, then combine each group. For instance: 3a + 4b – 2a + b. Rearrange as (3a – 2a) + (4b + b). Then compute: a + 5b.

较长的表达式可能包含多组同类项。最佳做法是重新排列各项，使同类项相邻，然后合并每一组。例如：3a + 4b – 2a + b。重排为 (3a – 2a) + (4b + b)，再计算得：a + 5b。

3a + 4b – 2a + b = (3a – 2a) + (4b + b) = a + 5b

Be careful not to drag the sign of a term when moving it. In the expression above, the term -2a carries the negative sign, so it moves with its minus sign.

移动项时要注意不要丢掉该项的符号。在上面的表达式中，-2a 带着负号，所以移动时它带着减号。

Example with more groups: 5x² – 3x + 2x² + 7x – 4. Group x² terms: 5x² + 2x² = 7x². Group x terms: -3x + 7x = 4x. The constant -4 stays alone. The simplified expression is 7x² + 4x – 4.

更多组的例子：5x² – 3x + 2x² + 7x – 4。合并 x² 项：5x² + 2x² = 7x²。合并 x 项：-3x + 7x = 4x。常数 -4 单独保留。化简后为 7x² + 4x – 4。

7. Common Mistakes to Avoid | 常见错误

Students often make mistakes when they confuse like terms with unlike terms. The most common error is adding x and x², treating them as the same. Remember that x is not x². Another common error is forgetting to include the sign when moving a term.

学生常常把同类项与非同类项混淆。最常见的错误是把 x 和 x² 当作同类项相加。请记住 x 不是 x²。另一个常见错误是在移动项时忘记包含其符号。

Mistake: 2x + 5x² = 7x² (incorrect). Correct: 2x + 5x² remains unchanged.
错误：2x + 5x² = 7x²（不正确）。正确：2x + 5x² 保持不变。

Mistake: 3a – 2a + 5 = a + 5a (incorrect). Correct: 3a – 2a = a, and the constant 5 remains, so a + 5.
错误：3a – 2a + 5 = a + 5a（不正确）。正确：3a – 2a = a，常数 5 保留，所以 a + 5。

Always double-check that every variable and exponent matches exactly before combining. If in doubt, write the expression with all terms expanded, such as 3 × x instead of 3x, to see the structure.

合并前务必确认每一个变量和指数完全匹配。如果有疑问，可以把表达式展开写，比如把 3x 写成 3 × x，以便看清结构。

8. Real-World Applications | 实际应用

Collecting like terms is not just an abstract exercise. It appears in geometry formulas, physics equations, and financial calculations. For example, the perimeter of a rectangle is often written as 2l + 2w. If the length is expressed in terms of x and the width in terms of x, you may need to combine them.

合并同类项不只是抽象的练习。它出现在几何公式、物理方程和财务计算中。例如，矩形的周长常写作 2l + 2w。如果长和宽都用 x 表示，你可能需要将同类项合并。

Suppose a rectangle has length 3x + 2 and width x – 1. The perimeter is 2(3x + 2) + 2(x – 1). Expand and collect: 6x + 4 + 2x – 2 = 8x + 2. This is a direct application of collecting like terms.

假设一个矩形的长是 3x + 2，宽是 x – 1。周长是 2(3x + 2) + 2(x – 1)。展开并合并：6x + 4 + 2x – 2 = 8x + 2。这就是合并同类项的直接应用。

In physics, if two forces are added, their components along the same axis are like terms. If F₁ = 3t + 5 and F₂ = 2t – 4, then the total force F = F₁ + F₂ = 5t + 1.

在物理中，如果将两个力相加，它们在相同轴上的分量就是同类项。若 F₁ = 3t + 5，F₂ = 2t – 4，则总力 F = F₁ + F₂ = 5t + 1。

9. Practice Problems with Solutions | 练习与解答

Try these exercises on your own, then check the solutions below.

请先自己尝试以下练习，然后核对下方解答。

Problem 1: Simplify 9x + 4x – 2x.
问题 1：化简 9x + 4x – 2x。

Solution: 9x + 4x – 2x = (9 + 4 – 2)x = 11x

解答：9x + 4x – 2x = (9 + 4 – 2)x = 11x

Problem 2: Simplify 2a + 3b – a + 5b.
问题 2：化简 2a + 3b – a + 5b。

Solution: (2a – a) + (3b + 5b) = a + 8b

解答：(2a – a) + (3b + 5b) = a + 8b

Problem 3: Simplify 4x² + 2x – x² + 7x.
问题 3：化简 4x² + 2x – x² + 7x。

Solution: (4x² – x²) + (2x + 7x) = 3x² + 9x

解答：(4x² – x²) + (2x + 7x) = 3x² + 9x

Problem 4: Simplify 6 – 3y + 5y – 2.
问题 4：化简 6 – 3y + 5y – 2。

Solution: (6 – 2) + (-3y + 5y) = 4 + 2y

解答：(6 – 2) + (-3y + 5y) = 4 + 2y

If you got the correct answers, you are ready for harder expressions. If not, review the steps and try again.

如果你做对了，说明你已准备好挑战更难的表达式。如果不对，请回顾步骤再试一次。

10. Summary and Key Takeaways | 总结与要点

Collecting like terms is a fundamental algebraic technique. To simplify any expression, identify the terms with identical variable parts, add or subtract their coefficients, and keep the variables unchanged. Never combine terms with different variables or different exponents.

合并同类项是一项基础代数技能。要化简任何表达式，请找出变量部分相同的项，将它们的系数相加或相减，并保持变量部分不变。绝不要合并变量不同或指数不同的项。

Like terms must have the same variable part and the same powers.
同类项必须具有相同的变量部分和相同的幂。
The coefficient includes the sign before the term.
系数包含该项前面的符号。
Simplify step by step, rearranging groups of like terms.
逐步化简，重排同类项的组。

Mastering this skill will help you solve equations, expand brackets, and handle more advanced topics like factorising and solving quadratics.

掌握这项技能将帮助你求解方程、展开括号，并处理更高级的内容，如因式分解和解二次方程。

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Collecting Like Terms | 合并同类项