Combining Like Terms in Algebra | 代数中的合并同类项

📚 Combining Like Terms in Algebra | 代数中的合并同类项

Algebra is like a language. When we write an expression such as ‘3x + 2x’, we are saying ‘three x’s plus two x’s’. Combining like terms is the process of making this language simpler by grouping terms that have exactly the same variable part.

代数像一门语言。当我们写出 ‘3x + 2x’ 这样的表达式时，我们其实在表示“三个 x 加两个 x”。合并同类项就是通过将变量部分完全相同的项归组，让这门语言更加简洁。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms whose variables, including their exponents, are identical. For example, 5x and -2x are like terms because both contain the single variable x. The numbers 7 and 3 are also like terms, because a constant term has no variable at all.

同类项是指变量部分（包括变量的指数）完全相同的项。例如，5x 和 -2x 是同类项，因为它们都含有单一变量 x。数字 7 和 3 也是同类项，因为常数项根本不含有变量。

2. Spotting Like Terms | 识别同类项

To spot like terms, look at the letters and their powers. The coefficient, the number in front, does not matter. For example, 4xy and 7xy are like terms, but 4x²y and 4xy are not like terms because the power of x is different.

要识别同类项，需要看字母及其幂指数，而系数（字母前面的数字）无关紧要。例如，4xy 和 7xy 是同类项，但 4x²y 和 4xy 不是同类项，因为 x 的幂不同。

Another example: 3a²b and 5a²b are like terms, while 3ab² and 5a²b are not like terms.

再如：3a²b 和 5a²b 是同类项，而 3ab² 和 5a²b 不是同类项。

3. The Golden Rule of Combining | 合并的基本法则

When combining like terms, add or subtract the coefficients but keep the variable part unchanged. If x is any variable and k and m are numbers, the rule may be written as kx + mx = (k + m)x.

合并同类项时，对系数进行加减，而变量部分保持不变。若 x 是任意变量，k 和 m 是数字，则法则可以写作 kx + mx = (k + m)x。

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

This rule also works for subtraction: 8y – 3y = 5y.

这条法则同样适用于减法：8y – 3y = 5y。

4. Working with Positive and Negative Coefficients | 处理正负系数

When the expression contains negative terms, think of each term as having a sign in front of it. For example, 7p – 4p + 2p means the coefficients are +7, -4 and +2. Add these together: 7 + (-4) + 2 = 5, so the result is 5p.

当表达式中含有负项时，把每一项看成前面带符号：7p – 4p + 2p 的系数分别是 +7、-4 和 +2。将它们相加：7 + (-4) + 2 = 5，所以结果是 5p。

Be careful with expressions like -5x – 3x. Both coefficients are negative, so the total is -8x.

要小心像 -5x – 3x 这样的表达式。两个系数都是负数，所以合并结果是 -8x。

5. Combining Terms with Several Variables | 合并多个变量

Terms can contain more than one variable. For example, combine 3ab + 2a – ab + 4a. The terms 3ab and -ab are like terms, and 2a and 4a are like terms. Group them separately: (3ab – ab) + (2a + 4a) = 2ab + 6a.

一个项可以含有多个变量。例如，合并 3ab + 2a – ab + 4a。其中 3ab 与 -ab 是同类项，2a 与 4a 是同类项。分别分组：(3ab – ab) + (2a + 4a) = 2ab + 6a。

You do not always need to write the answer in alphabetical order, but a consistent order helps avoid errors.

你不一定需要按字母顺序书写答案，但保持一致顺序有助于避免错误。

6. Combining After Expanding Brackets | 展开括号后合并

Often, you need to expand brackets before you can combine like terms. For example, simplify 2(x + 3) + 4x. First expand the bracket: 2x + 6 + 4x. Then combine like terms: 2x + 4x = 6x, so the final answer is 6x + 6.

通常，你需要先展开括号，然后才能合并同类项。例如，化简 2(x + 3) + 4x。先展开括号得 2x + 6 + 4x；再合并同类项：2x + 4x = 6x，所以最终答案为 6x + 6。

Remember: a negative sign outside a bracket changes every sign inside. Simplify 5 – (2x – 3). This becomes 5 – 2x + 3 = 8 – 2x.

要记住：括号外的负号会改变括号内每一项的符号。化简 5 – (2x – 3)，得到 5 – 2x + 3 = 8 – 2x。

7. Combining Fractional Coefficients | 合并分数系数

When coefficients are fractions, use the rules of fractions to add or subtract them. For example, ½x + ⅓x = (½ + ⅓)x = 5⁄6x.

当系数是分数时，使用分数的运算规则进行加减。例如，½x + ⅓x = (½ + ⅓)x = 5⁄6x。

Sometimes it is easiest to convert all fractions to a common denominator before combining.

有时，先将所有分数转换为公分母再进行合并会更简单。

8. Combining Powers and Exponents | 合并幂和指数

Like terms must have the same power of each variable. So x² and x are not like terms, and cannot be combined. However, 3x² + 5x² = 8x², because the variable part x² is identical.

同类项必须满足每个变量的幂相同。因此 x² 与 x 不是同类项，不能合并。但是 3x² + 5x² = 8x²，因为变量部分 x² 完全相同。

For higher powers, 2x³ + 4x³ – x³ = 5x³. But 2x³ + 4x² cannot be combined.

对于更高次幂，2x³ + 4x³ – x³ = 5x³。但 2x³ + 4x² 不能合并。

9. Common Mistakes to Avoid | 常见错误与避免方法

Do not add exponents when combining like terms. x² + x² = 2x², not x⁴.

合并同类项时不要将指数相加。x² + x² = 2x²，而不是 x⁴。
Do not combine x with x². They are unlike terms.

不要将 x 与 x² 合并。它们不是同类项。
Do not miss terms with coefficient 1. For example, x means 1x, so x + 3x = 4x.

不要忽略系数为 1 的项。例如，x 表示 1x，所以 x + 3x = 4x。
Do not forget the negative signs. Treat -2a as adding -2a, not subtracting 2a separately.

不要忽略负号。将 -2a 视为加上 -2a，而不是单独减去 2a。

10. Worked Examples for IGCSE | IGCSE 例题演练

Let us try typical IGCSE questions step by step.

让我们一步一步地练习典型 IGCSE 考题。

Example 1: Simplify 6m + 4n – 2m + 5n.

例 1：化简 6m + 4n – 2m + 5n。

Group the m terms and n terms: (6m – 2m) + (4n + 5n) = 4m + 9n.

将 m 项和 n 项分组：(6m – 2m) + (4n + 5n) = 4m + 9n。

Example 2: Simplify 4(2x – 1) + 3(x + 2).

例 2：化简 4(2x – 1) + 3(x + 2)。

Expand both brackets: 8x – 4 + 3x + 6. Combine like terms: 8x + 3x = 11x and -4 + 6 = 2. Answer: 11x + 2.

展开两个括号：8x – 4 + 3x + 6。合并同类项：8x + 3x = 11x，-4 + 6 = 2。答案为 11x + 2。

11. One More Challenge | 挑战练习

Try combining this expression: 2a²b – 3ab + a²b + 6ab – 4b.

尝试合并这个表达式：2a²b – 3ab + a²b + 6ab – 4b。

Collect the a²b terms: 2a²b + a²b = 3a²b. Collect the ab terms: -3ab + 6ab = 3ab. The term -4b has no like term. So the answer is 3a²b + 3ab – 4b.

收集 a²b 项：2a²b + a²b = 3a²b。收集 ab 项：-3ab + 6ab = 3ab。项 -4b 没有同类项。因此答案是 3a²b + 3ab – 4b。

12. Summary and Exam Tips | 总结与考试贴士

Combining like terms is a foundational skill in algebra. Always check the variable part carefully, keep the exponent unchanged, and add only the coefficients. Write each step clearly and avoid skipping too many steps in the exam.

合并同类项是代数的基础技能。始终仔细检查变量部分，保持指数不变，并且只对系数进行加减。在考试中，每步书写要清楚，不要跳步太多。

Remember: if you can substitute a simple value for the variable in both the original expression and your simplified answer, the two values should be equal. Use this to check your work.

请记住：如果你能在原表达式和化简后的表达式中代入同一个简单数值，两者的值应当相等。你可以用这个方法检查计算结果。

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Combining Like Terms in Algebra | 代数中的合并同类项