Simplifying Algebraic Expressions: Combining Like Terms | 代数式化简：合并同类项

📚 Simplifying Algebraic Expressions: Combining Like Terms | 代数式化简：合并同类项

In algebra, expressions often contain many terms. Simplifying them by combining like terms is a key skill for IGCSE Mathematics. It reduces clutter, makes patterns visible, and paves the way for solving equations.

在代数中，表达式经常包含许多项。通过合并同类项来化简它们，是 IGCSE 数学中的关键技能。这能使表达式更简洁、让规律更清晰，并为解方程打下基础。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part, including the same exponents. For example, 3x and 5x are like terms, while 3x and 3y are not. Similarly, 4x² and 2x² are like terms, but 4x² and 2x are not because the exponents differ.

同类项是指变量部分完全相同的项，包括指数也必须相同。例如，3x 和 5x 是同类项，而 3x 和 3y 不是。同样地，4x² 和 2x² 是同类项，但 4x² 和 2x 不是，因为指数不同。

Expression	Like Terms?	Reason
2a and 5a	Yes	Same variable a
4b and 4b²	No	Exponent differs
x²y and 3yx²	Yes	Same variables and exponents

2. Why Combine Like Terms? | 为什么要合并同类项？

Combining like terms simplifies an expression into its most compact form. This makes it easier to evaluate, substitute values, or solve equations. For instance, 5x + 3x can be written as 8x, which is much clearer for further calculations.

合并同类项能把表达式化简到最紧凑的形式。这使得代入数值、求值或解方程都更加容易。例如，5x + 3x 可以写成 8x，这在进行后续计算时清晰得多。

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

The same logic applies to terms with variables such as y, a, b, or expressions like xy.

同样的逻辑适用于 y、a、b 等变量，以及像 xy 这样的表达式。

3. Coefficients and the Invisible 1 | 系数与隐形系数 1

The coefficient is the number in front of a variable. When no number is shown, the coefficient is 1. For example, x means 1x. This is important when combining: x + 3x = 4x, because 1x + 3x = 4x.

系数是变量前面的数字。当没有写出数字时，系数就是 1。例如，x 表示 1x。这在合并时很重要：x + 3x = 4x，因为 1x + 3x = 4x。

x + 3x = (1 + 3)x = 4x

Similarly, -x means -1x. So -x + 4x = 3x.

类似地，-x 表示 -1x。所以 -x + 4x = 3x。

4. Using the Commutative Property to Group | 利用交换律进行分组

The commutative property of addition allows us to reorder terms without changing the result. We use this to bring like terms together before combining them.

加法的交换律允许我们在不改变结果的前提下重新排列各项。我们利用这一点把同类项放在一起，然后再合并。

3x + 2y – x + 5y = (3x – x) + (2y + 5y) = 2x + 7y

Notice that the sign of each term moves with it. In the example, -x is subtracted, so we keep the negative sign when grouping.

注意每一项的符号都要跟着它一起移动。在示例中，-x 是做减法，所以分组时我们要保留负号。

5. Combining Terms with Different Exponents | 合并不同指数的项

Terms with different exponents are not like terms. For example, x² and x cannot be combined into a single x-term. You must keep them separate.

指数不同的项不是同类项。例如，x² 和 x 不能合并成一个含 x 的项。你必须把它们分开保留。

2x² + 3x – x² + x = (2x² – x²) + (3x + x) = x² + 4x

Here, x² and x are treated as separate families. Even when simplifying, the degrees do not merge.

这里，x² 和 x 被认为是不同的“家族”。即使在化简时，次数也不会合并。

6. Handling Negative Coefficients and Signs | 处理负系数与符号

When combining terms, subtraction can lead to negative coefficients. For example, 4a – 6a = -2a. Always pay attention to the sign of each coefficient.

合并项时，减法可能导致负系数。例如，4a – 6a = -2a。始终要注意每个系数的正负号。

7b – 9b = (7 – 9)b = -2b
4c – (-3c) = 4c + 3c = 7c

Remember that subtracting a negative is the same as adding a positive.

记住：减去一个负数等于加上一个正数。

7. Simplifying Expressions with Multiple Variables | 含多个变量的表达式化简

Expressions often contain more than one variable. Group and combine each variable family separately.

表达式中常常包含多个变量。分别对每个变量家族进行分组和合并。

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

Constant numbers (without variables) can also be combined together. For instance, 4 + 7 = 11 in an expression.

常数（不含变量的数字）也可以合并在一起。例如，表达式中 4 + 7 = 11。

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

8. Combining Like Terms in Word Problems | 应用题中的合并同类项

Word problems often generate algebraic expressions that need simplifying. For example, the perimeter of a triangle with sides 2x, 3x + 1, and x – 2 is found by adding the three sides.

应用题常常产生需要化简的代数表达式。例如，一个三角形的三边长分别为 2x、3x + 1 和 x – 2，其周长等于三边之和。

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

This simplified expression gives the perimeter in terms of x, and can be evaluated for any given value of x.

化简后的表达式给出了用 x 表示的周长，对于任意给定的 x 值都可以直接求值。

9. Common Mistakes to Avoid | 常见错误避免

Combining coefficients without checking that the variable parts are identical.

不检查变量部分是否相同就合并系数。
Adding exponents when combining terms. Remember: x² + x² = 2x², not x⁴.

合并项时错误地相加指数。记住：x² + x² = 2x²，而不是 x⁴。
Forgetting the invisible coefficient 1, especially with negative signs: -x is -1x.

忘记隐形系数 1，尤其是在负数符号中：-x 就是 -1x。
Losing signs when rearranging terms. Always keep the sign attached to the term.

重新排列项时丢失符号。始终让符号跟随它所对应的项。

10. Worked Examples and Practice | 例题详解与练习

Work through this example step by step:

请逐步完成这个例子：

4a – 2b + 3a + 5b – b

Group like terms: (4a + 3a) + (-2b + 5b – b).

分组同类项：(4a + 3a) + (-2b + 5b – b)。
Combine each group: 7a + ( -2 + 5 – 1)b = 7a + 2b.

合并每组：7a + ( -2 + 5 – 1)b = 7a + 2b。

Try yourself: 5x² + 3x – 2x² + x – 4x. Which terms are like? The correct simplified form is 3x² + 0x, or simply 3x².

自己尝试：5x² + 3x – 2x² + x – 4x。哪些是同类项？正确化简形式是 3x² + 0x，也就是 3x²。

11. Summary and Exam Tips | 总结与考试要点

To combine like terms successfully: identify terms with the same variable and exponent, group them using the commutative property, then add or subtract their coefficients. Keep signs attached and do not change exponents.

要成功合并同类项：找出具有相同变量和指数的项，利用交换律分组，然后对它们的系数做加法或减法。保持符号跟随，并且不要改变指数。

In IGCSE exams, always show the grouping step in your working. This makes it easier to catch sign errors and earns method marks.

在 IGCSE 考试中，一定要在步骤中展示分组过程。这样更容易发现符号错误，也能获得方法分。

Practice daily with a variety of expressions, especially those with negatives and multiple variables. This skill will become automatic and form a solid foundation for later algebra.

每天练习各种表达式，尤其是含负数与多个变量的表达式。这项技能会变得娴熟，并为后续代数学习打下坚实基础。

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Simplifying Algebraic Expressions: Combining Like Terms | 代数式化简：合并同类项