Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, the ability to combine like terms (also called collecting like terms) is a fundamental skill. It appears in almost every examination, from simplifying expressions to solving equations. This article will guide you through the definition, rules, and common pitfalls, with plenty of worked examples.

在代数中，合并同类项（collecting like terms）是一项基础技能。从化简表达式到解方程，几乎每场考试都会用到。本文将系统讲解其定义、规则、常见错误以及大量例题，帮助你熟练掌握这一考点。

1. What Is an Algebraic Expression? | 什么是代数表达式？

An algebraic expression is a collection of numbers, variables, and operations such as addition and subtraction. For example, 3x + 5y − 2 is an algebraic expression. It contains no equals sign, so it is not an equation.

代数表达式是由数字、变量和运算符号组成的集合。例如，3x + 5y − 2 就是一个代数表达式。它不包含等号，因此不是方程。

Expressions can be simplified by combining like terms. This makes the expression shorter and easier to work with in equations, graphs, and word problems.

表达式可以通过合并同类项来化简。化简后的表达式更简洁，更容易在方程、图像和应用题中使用。

2. Breaking Down Terms | 多项式的项

A term is a number, a variable, or a product of numbers and variables. In the expression 3x + 5y − 2, the terms are 3x, 5y, and −2.

项（term）是指一个数、一个变量，或者数字与变量的乘积。在表达式 3x + 5y − 2 中，项是 3x、5y 和 −2。

Each term has two parts: a numerical coefficient and a variable part. A constant term has no variable.

每一项有两个部分：数字系数（coefficient）和变量部分。常数项不含变量。

Term	Coefficient	Variable part	Constant?
3x	3	x	No
5y	5	y	No
−2	−	−	Yes

In the term −2, the coefficient is the whole number −2 itself, and the variable part is empty.

在项 −2 中，系数就是 −2，它没有变量部分。

3. Like and Unlike Terms | 同类项与非同类项

Like terms are terms that have exactly the same variable part, including the same power. For example, 3x and 5x are like terms, because both contain the variable x to the power 1.

同类项是指变量部分完全相同的项，包括相同的指数。例如，3x 和 5x 是同类项，因为它们的变量都是 x 的一次方。

Similarly, 2x² and 7x² are like terms. However, 2x² and 4x are not like terms, because the powers of x are different.

同样，2x² 和 7x² 是同类项。但是，2x² 与 4x 不是同类项，因为 x 的指数不同。

Terms with different variables, such as 3x and 3y, are also not like terms.

变量不同的项，如 3x 和 3y，也不是同类项。

Expression	Like terms?
4x and 9x	Yes
4x and 4y	No
5x² and 2x²	Yes
5x² and 2x	No
3xy and 7xy	Yes

4. The Rule of Combining | 合并的规则

To combine like terms, add or subtract their coefficients and keep the variable part exactly the same.

合并同类项时，将系数相加或相减，同时保持变量部分完全不变。

A simple example is 3x + 5x. Since both terms have the same variable x, we add the coefficients 3 and 5 to get 8, and keep x unchanged.

一个简单例子是 3x + 5x。因为两项都含有变量 x，我们将系数 3 和 5 相加得到 8，并保持 x 不变。

3x + 5x = 8x

In general, for any value of a and b:

一般地，对于任意 a 和 b：

ax + bx = (a + b)x

This rule also works for subtraction: 11y − 4y = 7y.

减法同样适用：11y − 4y = 7y。

5. Combining Terms with Powers | 含指数的项合并

When an expression contains different powers of the same variable, we cannot combine them as one. We must group x² terms separately from x terms.

当表达式包含同一个变量的不同次幂时，不能将它们合并成一项。我们必须把 x² 项与 x 项分开归类。

For example, simplify 2x² + 3x + 4x² + x.

例如，化简 2x² + 3x + 4x² + x。

Group the like terms: (2x² + 4x²) + (3x + x).

将同类项分组：(2x² + 4x²) + (3x + x)。

Then combine them:

然后合并：

(2 + 4)x² + (3 + 1)x = 6x² + 4x

Notice that 6x² and 4x remain separate terms, because they have different powers.

注意，6x² 和 4x 仍然是分开的项，因为它们的指数不同。

6. Combining Multiple Variables | 多变量的合并

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

变量不同的项不能合并。例如，2x + 3y 不能再化简，因为 x 和 y 是不同的变量。

However, terms with the same variable pair, such as 2xy and 5xy, are like terms. We combine them by adding the coefficients:

但变量对相同的项，如 2xy 和 5xy，就是同类项。我们通过相加系数来合并：

2xy + 5xy = 7xy

It is also useful to remember that xy and yx are the same term, because multiplication is commutative.

还需要记住，xy 与 yx 是相同的项，因为乘法满足交换律。

For terms like 2x²y and 7x²y, they are like because they both contain x²y. We can simplify 2x²y + 7x²y = 9x²y.

像 2x²y 和 7x²y 这样的项，因为都含有 x²y，所以是同类项。我们可以化简 2x²y + 7x²y = 9x²y。

7. Simplifying after Expanding | 展开后再合并

Many exam questions require you to expand brackets first and then combine like terms. This is a two-step process.

许多考试题目要求先展开括号，再合并同类项。这是两步过程。

Simplify 2(x + 3) + 3(x − 1).

化简 2(x + 3) + 3(x − 1)。

First expand each bracket:

首先展开每个括号：

2x + 6 + 3x − 3

Now combine the like terms:

现在合并同类项：

(2x + 3x) + (6 − 3) = 5x + 3

Write the final simplified expression as 5x + 3.

最终化简结果为 5x + 3。

8. Handling Negative Coefficients | 处理负系数

When combining like terms, always treat the sign in front of each term as part of the coefficient.

合并同类项时，应始终将每一项前面的符号视为系数的一部分。

For example, simplify 5x − 3x + 2. Here, the coefficient of the second x term is −3.

例如，化简 5x − 3x + 2。这里，第二项 x 的系数是 −3。

So we calculate 5 − 3 = 2, and the result is 2x + 2.

因此我们计算 5 − 3 = 2，结果为 2x + 2。

Be careful when subtracting a bracket. For example, simplify 2(3x − 1) − (x + 2).

减去括号时要格外小心。例如，化简 2(3x − 1) − (x + 2)。

First expand: 6x − 2 − x − 2. Note that −(x + 2) becomes −x − 2.

先展开：6x − 2 − x − 2。注意 −(x + 2) 变为 −x − 2。

Now combine: (6x − x) + (−2 − 2) = 5x − 4.

然后合并：(6x − x) + (−2 − 2) = 5x − 4。

9. Common Errors | 常见错误

Here are some frequent mistakes made by students when combining like terms.

以下是学生在合并同类项时常见的错误。

Combining different variables: 3x + 2y is not 5xy. They cannot be combined.

合并不同变量：3x + 2y 不等于 5xy，它们是不同变量，不能合并。
Combining different powers: 2x² + 3x is not 5x². The powers must match.

合并不同指数：2x² + 3x 不等于 5x²，指数必须相同。
Forgetting the negative sign: 4x − 7x = −3x, not 3x or 11x.

忘记负号：4x − 7x = −3x，而不是 3x 或 11x。
Sign errors when removing brackets: −(x + 3) is −x − 3, not −x + 3.

去括号时符号错误：−(x + 3) 等于 −x − 3，而不是 −x + 3。

10. Real-World Application | 实际应用

Combining like terms is used when calculating the perimeter of a shape. Suppose a rectangle has width (2x + 1) and length (3x + 2).

在求几何图形的周长时，会用到合并同类项。假设一个长方形的宽是 (2x + 1)，长是 (3x + 2)。

The perimeter is twice the sum of length and width:

周长是长与宽之和的两倍：

P = 2[(3x + 2) + (2x + 1)]

Simplify inside the bracket first: 3x + 2x + 2 + 1 = 5x + 3.

先化简括号内：3x + 2x + 2 + 1 = 5x + 3。

Then multiply by 2: P = 2(5x + 3) = 10x + 6.

然后乘以 2：P = 2(5x + 3) = 10x + 6。

This result tells you the perimeter in terms of x, which can be used to solve for x if the perimeter is given.

这个结果

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Combining Like Terms | 合并同类项