Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, an expression often contains several terms that share the same variable part. Combining like terms is the process of merging these terms into a single term, which makes the expression shorter and easier to work with.

在代数中，一个表达式常常包含若干具有相同变量部分的项。合并同类项就是将这样的项合并成一项的过程，它让表达式更加简洁，也更容易进一步运算。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable factors and the same exponents. Their coefficients may be different.

同类项指含有完全相同的变量以及相同指数的项。它们的系数可以不同。

For example, 3x and 5x are like terms because both have the variable x with exponent 1. Similarly, 2x² and -7x² are like terms because both contain x².

例如，3x 和 5x 是同类项，因为它们都含有一次的变量 x。同样地，2x² 和 -7x² 也是同类项，因为它们都含有 x²。

Constant terms, such as 4 and -9, are also considered like terms because neither contains a variable.

常数项，例如 4 和 -9，也被认为是同类项，因为它们都不含变量。

Like: 7a and 3a
Like: 4x² and -x²
Not like: 5x and 5x²
Not like: 2xy and 3x

同类：7a 与 3a
同类：4x² 与 -x²
不同类：5x 与 5x²
不同类：2xy 与 3x

2. Why Combine Like Terms? | 为什么合并同类项

Combining like terms transforms a long, cluttered expression into a compact form. This is especially useful before solving equations or substituting values.

合并同类项能把冗长杂乱的表达式变为紧凑的形式。这在解方程或代入数值之前尤其有用。

For example, 4x + 3x – 2x can be simplified to 5x. This simplification makes it easier to see the overall coefficient of x.

例如，4x + 3x – 2x 可化简为 5x。这种化简让我们更容易看出 x 的总系数。

In exams, leaving an expression unsimplified can lose marks, so this skill is essential for IGCSE Mathematics.

在考试中，若不化简表达式可能会被扣分，因此这一技能对 IGCSE 数学至关重要。

3. The Distributive Law Behind Combining | 合并背后的分配律

Combining like terms is really an application of the distributive law in reverse. When we write 3x + 5x, we can see x as a common factor:

合并同类项实际上是把分配律反过来运用。当我们看到 3x + 5x 时，可以把 x 看作公因式：

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

The same principle works for any variable or combination of variables:

同样的原理适用于任何变量或变量组合：

2ab – 7ab = (2 – 7)ab = -5ab

This idea is the foundation of all simplification in algebra.

这一思想是代数化简的根本基础。

4. Combining Positive and Negative Coefficients | 合并正系数与负系数

When merging terms, pay close attention to the sign in front of each coefficient. A minus sign belongs to the term that follows it.

合并时，要密切关注每一项前面的正负号。负号属于它后面的那一项。

Consider the expression 6a – 9a + 4a. First identify the coefficients: +6, -9 and +4.

考虑表达式 6a – 9a + 4a。首先确定系数：+6、-9 和 +4。

6a – 9a + 4a = (6 – 9 + 4)a = 1a = a

Notice that when the final coefficient is 1, we usually write just the variable letter.

注意，当最终系数为 1 时，我们通常只写变量字母。

For negative results, the minus sign must be kept. For example:

如果结果是负的，负号必须保留。例如：

2p – 5p = -3p

5. Terms with Different Variable Powers | 变量幂指数不同的项

Only terms with the exact same variable power can be combined. An x term cannot be merged with an x² term.

只有变量幂指数完全相同的项才能合并。一次 x 项不能与 x² 项合并。

For example, in the expression 3x² + 2x + 5x² – 4x, we can combine the x² terms and the x terms separately:

例如，在表达式 3x² + 2x + 5x² – 4x 中，我们可以分别合并 x² 项和 x 项：

3x² + 5x² = 8x²
2x – 4x = -2x

So the simplified expression is 8x² – 2x.

因此化简结果为 8x² – 2x。

Just because both terms contain x does not mean they are like terms. The exponent must match.

仅因为两项都含有 x 并不意味着它们是同类项。指数必须相同。

6. Simplifying Expressions with Brackets | 括号表达式的化简

When an expression contains brackets, expand the brackets first, then combine like terms. This is a very common IGCSE question type.

当表达式中含有括号时，应先去括号，再合并同类项。这是 IGCSE 中非常常见的题型。

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

示例：化简 3(x + 2) + 4x。

First expand: 3(x + 2) = 3x + 6.

先去括号：3(x + 2) = 3x + 6。

Then combine with 4x:

然后与 4x 合并：

3x + 6 + 4x = (3x + 4x) + 6 = 7x + 6

If there is a minus sign before a bracket, remember to distribute it carefully:

如果括号前是减号，注意要仔细分配：

5y – 2(3y – 1) = 5y – 6y + 2 = -y + 2

7. Combining Terms with Multiple Letters | 含多个字母的项的合并

Terms may contain more than one variable, such as xy, x²y or ab². For these to be like terms, the entire variable product must match exactly.

项可能含有多个变量，例如 xy、x²y 或 ab²。要成为同类项，整个变量乘积必须完全一致。

For instance, 3xy and -7xy are like terms because both have xy. They can be combined:

例如，3xy 和 -7xy 是同类项，因为它们都含有 xy。它们可以合并：

3xy – 7xy = -4xy

However, x²y and xy² are not like terms because the powers of x and y are arranged differently.

然而，x²y 和 xy² 不是同类项，因为 x 和 y 的幂排列不同。

Can combine: 2x²y + 5x²y = 7x²y
Cannot combine: 2x²y and 5xy²

可合并：2x²y + 5x²y = 7x²y
不可合并：2x²y 与 5xy²

8. Real-World Applications | 实际应用

Simplifying expressions is useful in real contexts, such as finding the perimeter of a shape with algebraic side lengths.

化简表达式在实际问题中很有用，例如求具有代数边长的图形的周长。

Example: A rectangle has length x + 4 and width 2x – 1. Its perimeter is:

示例：一个矩形的长为 x + 4，宽为 2x – 1。其周长为：

2(x + 4) + 2(2x – 1) = 2x + 8 + 4x – 2 = 6x + 6

This result makes it easy to evaluate the perimeter for any value of x.

这个结果让我们能很容易地对任意 x 值计算周长。

Similar techniques appear in areas such as physics formulas, economics, and computer science.

类似技巧也出现在物理公式、经济学和计算机科学等领域。

9. Common Mistakes | 常见错误

Students often make mistakes when the signs are negative or when the exponent is not considered carefully.

学生常在符号为负或指数未被仔细考虑时犯错。

Mistake 1: Writing 2x + 3 = 5x. This is wrong because 2x and 3 are not like terms.
Mistake 2: Writing 3x – x² = 2x. The powers are different, so they cannot be merged.
Mistake 3: Dropping a negative sign: 4a – 6a + 2a = 0a. The correct result is 0, but writing 0a is not simplified.
Mistake 4: Forgetting to apply a sign to all terms after a minus bracket.

错误一：将 2x + 3 写成 5x。这是错的，因为 2x 和 3 不是同类项。
错误二：将 3x – x² 写成 2x。幂不同，不能合并。
错误三：丢掉负号：4a – 6a + 2a = 0a。正确结果是 0，但写 0a 不是最简形式。
错误四：减号括号后忘了给所有项变号。

To avoid these errors, always underline or group the like terms before combining them.

为避免这些错误，合并前先给同类项画线或分组。

10. Exam Tips and Practice | 考试技巧与练习

In the exam, always show your working line by line. Even if the final answer is wrong, you may receive method marks.

考试中，一定要逐步写出过程。即使最终答案错了，也可能获得步骤分。

Check your answer by substituting a simple value, such as x = 1, into the original and simplified expressions. They should give the same result.

检查答案时，可将一个简单值如 x = 1 分别代入原表达式和化简后的表达式，结果应相同。

Try these practice problems:

请尝试以下练习题：

Simplify: 7a + 3a – 10a
Simplify: 5x² + 2x – 3x² – x
Simplify: 2(3n + 4) – 5n
Simplify: 4ab – 2a + 3ab + 7a

化简：7a + 3a – 10a
化简：5x² + 2x – 3x² – x
化简：2(3n + 4) – 5n
化简：4ab – 2a + 3ab + 7a

Answers: 1. 0; 2. 2x² + x; 3. n + 8; 4. 7ab + 5a.

答案：1. 0；2. 2x² + x；3. n + 8；4. 7ab + 5a。

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Combining Like Terms | 合并同类项