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Combining Like Terms | 合并同类项

2026-08-21 19:09 · 本文部分内容由AI辅助创作

📚 Combining Like Terms | 合并同类项

In IGCSE Mathematics, simplifying algebraic expressions is one of the first essential skills. Combining like terms means gathering terms that have exactly the same variable part and then adding or subtracting their coefficients.

在 IGCSE 数学中，化简代数表达式是最基本的关键技能之一。合并同类项是指把含有完全相同变量部分的项集中起来，并对其系数进行加减运算。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms whose variable parts are exactly the same. The variables must match in letter and in power. For example, 3x and −5x are like terms because both contain x to the power 1.

同类项是指变量部分完全相同的项。变量必须在字母和次数上都一致。例如，3x 和 −5x 是同类项，因为它们都含有 x 的一次方。

Constants are also like terms with each other. For example, 7 and −2 are like terms because neither has a variable part. However, 3x and 3x² are not like terms, because the powers of x are different.

常数项之间也互为同类项。例如，7 和 −2 是同类项，因为它们都没有变量部分。但是，3x 和 3x² 不是同类项，因为 x 的次数不同。

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

When an expression contains many terms, it is easier to evaluate, compare, or use in further algebra if it is written in its simplest form. Combining like terms reduces the number of terms and makes the expression clearer.

当一个表达式含有许多项时，如果能把它写成最简形式，就会更容易代入求值、比较或继续运算。合并同类项可以减少项数，使表达式更加清晰。

For example, 2x + 3x + 4x may look like three separate pieces, but it is simpler to write it as 9x. This saves time and reduces mistakes in solving equations or substituting values.

例如，2x + 3x + 4x 看起来是三个独立的部分，但更简单的写法是 9x。这在解方程或代入数值时可以节省时间并减少错误。

3. Identifying Like Terms | 识别同类项

To identify like terms, ignore the coefficient and focus on the variable part. The variable part includes the letters and their powers. If two terms have the same variable part, they can be combined.

要识别同类项，可以先忽略系数，只关注变量部分。变量部分包括字母及其次数。如果两项的变量部分相同，它们就可以合并。

For example, in 7a²b and −3a²b, the variable part is a²b in both terms. The coefficients 7 and −3 do not affect whether the terms are like terms.

例如，在 7a²b 和 −3a²b 中，两项的变量部分都是 a²b。系数 7 和 −3 不会影响它们是否为同类项。

4. The Basic Rule: Add or Subtract Coefficients | 基本法则：系数相加减

When combining like terms, only the coefficients are added or subtracted. The variable part stays exactly the same. This is because the variable part is a common factor in all like terms.

合并同类项时，只对系数进行加减运算，变量部分保持不变。这是因为变量部分是所有同类项的公因式。

al, for like terms ax and bx, the result is (a + b)x. For subtraction, ax − bx = (a − b)x.

一般来说，对于同类项 ax 和 bx，合并结果是 (a + b)x。对于减法，ax − bx = (a − b)x。

ax + bx = (a + b)x

ax − bx = (a − b)x

This rule applies no matter how many like terms are present. You simply add up all the numerical coefficients and keep the variable part unchanged.

无论有多少个同类项，这个法则都适用。你只需要把所有数字系数相加，变量部分保持不变。

5. Single-Variable Examples | 单变量示例

ave the same variable part x, so we add the coefficients: 3 + 5 − 2 = 6. The answer is 6x.

化简 3x + 5x − 2x。这三项的变量部分都是 x，所以把系数相加：3 + 5 − 2 = 6。答案是 6x。

3x + 5x − 2x = (3 + 5 − 2)x = 6x

the coefficient of the last term is 1. The result is 10 − 3 + 1 = 8, so the answer is 8a.

化简 10a − 3a + a。记住 a 表示 1a，所以最后一项的系数是 1。计算结果是 10 − 3 + 1 = 8，因此答案是 8a。

10a − 3a + a = (10 − 3 + 1)a = 8a

6. Combining Terms with Powers | 含幂次的项合并

Terms with different powers cannot be combined. For example, x and x² are not like terms, so 3x + 4x² must be left as it is. You can only combine x² with x² and x with x.

次数不同的项不能合并。例如，x 和 x² 不是同类项，所以 3x + 4x² 必须保持原样。你只能把 x² 与 x² 合并，把 x 与 x 合并。

iving 5x². Then combine the x terms: 5 − 7 = −2, giving −2x. The final answer is 5x² − 2x.

化简 2x² + 5x + 3x² − 7x。先合并 x² 项：2 + 3 = 5，得到 5x²。再合并 x 项：5 − 7 = −2，得到 −2x。最终答案是 5x² − 2x。

2x² + 5x + 3x² − 7x = 5x² − 2x

Always keep the power exactly the same when combining. Adding coefficients does not change the exponent of the variable.

合并时一定要保持次数完全不变。系数相加不会改变变量的指数。

7. Expressions with Multiple Variables | 多变量表达式

In expressions with more than one variable, each distinct variable part must be handled separately. Terms like 4ab and −2ab are like terms, but 4ab and 3a are not.

在含有多个变量的表达式中，不同的变量部分必须分别处理。像 4ab 和 −2ab 这样的项是同类项，但 4ab 和 3a 不是同类项。

4 − 2 = 2, giving 2ab. Combine the a terms: 3 + 6 = 9, giving 9a. The answer is 2ab + 9a.

化简 4ab + 3a − 2ab + 6a。合并 ab 项：4 − 2 = 2，得到 2ab。合并 a 项：3 + 6 = 9，得到 9a。答案是 2ab + 9a。

4ab + 3a − 2ab + 6a = 2ab + 9a

Be careful with terms such as xy and x²y. They are not like terms because the power of x is different. Similarly, ab and a²b are not like terms.

注意 xy 和 x²y 这样的项。它们不是同类项，因为 x 的次数不同。同样，ab 和 a²b 也不是同类项。

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8. Constants and Mixed Terms | 常数与混合项处理

Constant terms can always be combined with each other, no matter what variables appear elsewhere in the expression. For example, in 2x + 5 − 3x + 8, the constants 5 and 8 can be combined to make 13.

常数项总是可以互相合并，无论表达式中的其他变量是什么。例如，在 2x + 5 − 3x + 8 中，常数 5 和 8 可以合并成 13。

− 3 = −1, giving −x. Combine the constants: 5 + 8 = 13. The answer is −x + 13, or 13 − x.

化简 2x + 5 − 3x + 8。合并 x 项：2 − 3 = −1，得到 −x。合并常数项：5 + 8 = 13。答案是 −x + 13，也可以写成 13 − x。

2x + 5 − 3x + 8 = −x + 13

It is often conventional to write the variable terms before the constant term, but both orders are mathematically correct. Choose the form your teacher or examiner expects.

通常习惯把变量项写在常数项前面，但两种顺序在数学上都是正确的。选择老师或考官要求的形式即可。

9. Brackets Before Combining | 先去括号再合并

When an expression contains brackets, you must expand the brackets first. Use the distributive law to multiply the term outside by each term inside the brackets, then combine like terms.

当表达式中含有括号时，必须先展开括号。使用分配律，将括号外的项乘以括号内的每一项，然后再合并同类项。

the x terms: 3 + 2 = 5, giving 5x. Combine the constants: 6 − 2 = 4. The answer is 5x + 4.

化简 3(x + 2) + 2(x − 1)。先展开：3x + 6 + 2x − 2。然后合并 x 项：3 + 2 = 5，得到 5x。合并常数项：6 − 2 = 4。答案是 5x + 4。

3(x + 2) + 2(x − 1) = 3x + 6 + 2x − 2 = 5x + 4

negative sign means multiply every term inside by −1, so it becomes 4x − 2x − 3 = 2x − 3.

当括号前有负号时要特别小心。化简 4x − (2x + 3)。负号表示将括号内的每一项都乘以 −1，因此变为 4x − 2x − 3 = 2x − 3。

4x − (2x + 3) = 4x − 2x − 3 = 2x − 3

10. Common Mistakes to Avoid | 常见错误

One common mistake is trying to combine unlike terms. For example, 3x + 2x² cannot be simplified to 5x² or 5x. The terms must have exactly the same variable part.

一个常见错误是试图合并不是同类项的项。例如，3x + 2x² 不能化简为 5x² 或 5x。各项的变量部分必须完全相同。

of 1. The term x is the same as 1x, so x + 2x = 3x, not 2x or x². Similarly, −x means −1x.

另一个错误是忘记隐藏的系数 1。x 与 1x 相同，所以 x + 2x = 3x，而不是 2x 或 x²。同样，−x 表示 −1x。

x = 2x, not x². The exponent only changes when you multiply powers, not when you add them.

学生在相加时有时还会改变变量的次数。请记住，x + x = 2x，而不是 x²。指数只有在幂相乘时才会改变，相加时不会改变。

11. Exam-Style Practice Questions | 考试型练习题

Try these exam-style questions before checking the answers. Simplify each expression fully.

在查看答案之前，先尝试完成这些考试型练习题。请将每个表达式化简到最简形式。

Answers: The first expression simplifies to 5a + 8b. The second becomes 4x² + 7x. The third expands to 10x + 5 − 3x + 6, which gives 7x + 11. The fourth becomes 2xy + 4y.

答案：第一个表达式化简为 5a + 8b。第二个变为 4x² + 7x。第三个展开为 10x + 5 − 3x + 6，结果为 7x + 11。第四个变为 2xy + 4y。

When completing exam questions, always reorder the expression if needed, but keep the sign in front of each term. Writing the steps clearly helps you avoid losing marks.

在完成考试题目时，如果需要可以重排表达式，但要保留每一项前面的符号。清晰地写出步骤有助于避免失分。

12. Summary Checklist | 总结清单

To combine like terms successfully, first identify terms with exactly the same variable part and power. Then add or subtract their coefficients while keeping the variable part unchanged.

要成功合并同类项，首先要识别变量部分和次数完全相同的项。然后将它们的系数相加或相减，同时保持变量部分不变。

Always expand brackets before combining terms, and be careful with negative signs in front of brackets. Constants are combined separately from variable terms.

在合并之前一定要展开括号，并小心括号前的负号。常数项要与变量项分开合并。

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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