Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, one of the most fundamental skills is simplifying expressions by collecting like terms. This process makes expressions easier to read and prepares you for solving equations and working with formulas.

在代数中，最基本的技能之一就是通过合并同类项来化简表达式。这个过程使表达式更易于阅读，也为解方程和运用公式打下基础。

1. What Are Terms and Like Terms? | 什么是项和同类项？

A term is a single number, a variable, or a product of numbers and variables. For example, in the expression 4x + 7 – 3x, the terms are 4x, 7, and -3x.

项是一个单独的数、一个变量，或者数与变量的乘积。例如，在表达式 4x + 7 – 3x 中，项是 4x、7 和 -3x。

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

同类项是指含有完全相同变量因式且指数也相同的项。例如，5x 和 3x 是同类项，因为它们都含有一次方的 x。

Here is a quick guide to identify like terms:

以下是快速识别同类项的方法：

Expression	Like terms?	Explanation
2x and 5x	Yes	Same variable x, same exponent 1.
2x and 5x²	No	Exponents differ (1 vs 2).
4xy and 7yx	Yes	Order does not matter: xy = yx.
4xy and 7x	No	Different variable factors.

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

Combining like terms simplifies an expression so that it has fewer terms. This makes the expression clearer and easier to substitute values into.

合并同类项可以化简表达式，使项数更少。这能让表达式更清晰，也更容易代入数值。

For example, the expression 3x + 5x can be written as 8x. Both forms are equal, but 8x is simpler and faster to use in further calculations.

例如，表达式 3x + 5x 可以写成 8x。两种形式是相等的，但 8x 更简单，在后续计算中使用更快。

Simplified expressions are also required when solving equations. If you leave 3x + 5x uncombined on one side, you may still solve it, but it is much tidier to combine first.

化简表达式也是解方程时的必要步骤。如果你在一边保留 3x + 5x 而不合并，仍然可以解方程，但先合并会更整洁。

3. The Basic Rule: Add Coefficients, Keep Variables | 基本规则：系数相加，变量不变

To combine like terms, add or subtract the numerical coefficients, and keep the variable part unchanged. The variable acts as a common factor.

合并同类项时，将数字系数相加或相减，变量部分保持不变。变量作为共同的因式。

For example:

例如：

4x + 7x = (4 + 7)x = 11x

The coefficient 4 and 7 are added, while x remains the same.

系数 4 和 7 相加，而 x 保持不变。

This works because of the distributive property: 4x + 7x = x(4 + 7) = 11x.

这是因为乘法分配律：4x + 7x = x(4 + 7) = 11x。

Consider another example with subtraction:

再看一个含减法的例子：

9y – 4y = (9 – 4)y = 5y

Always combine only the numbers in front of the same variable.

始终只合并相同变量前面的数字。

4. Combining Positive and Negative Terms | 正负项的合并

Like terms can have positive or negative coefficients. You must pay attention to the sign in front of each term.

同类项的系数可以是正数或负数。你必须注意每一项前面的符号。

Consider the expression 6x – 3x + 2x. Work from left to right:

考虑表达式 6x – 3x + 2x。从左向右计算：

6x – 3x = 3x, then 3x + 2x = 5x

So 6x – 3x + 2x simplifies to 5x.

所以 6x – 3x + 2x 化简为 5x。

When there are several terms, it helps to group them by sign, or use the idea of adding directed numbers.

当项数较多时，可以按正负分组，或者利用有向数的加法来处理。

Example: 2a – 5a + 8a – a. Collect the coefficients: 2 – 5 + 8 – 1 = 4. Therefore the expression is 4a.

例：2a – 5a + 8a – a。合并系数：2 – 5 + 8 – 1 = 4。因此表达式为 4a。

Be careful: a term such as -a has coefficient -1.

注意：像 -a 这样的项，其系数是 -1。

5. Working with Different Variables | 不同变量怎么办？

Terms with different variables are not like terms, so they cannot be combined. For example, x and y cannot be added to form a single term.

含有不同变量的项不是同类项，因此不能合并。例如，x 和 y 不能相加成一个单独的项。

The expression 3x + 2y is already in its simplest form. We cannot write it as 5xy or 5x.

表达式 3x + 2y 已经是最简形式。我们不能把它写成 5xy 或 5x。

Sometimes an expression contains several groups of like terms. You must combine each group separately.

有时一个表达式包含多个同类项组。你必须分别合并每一组。

Example: 4x + 3y – x + 2y. First combine the x terms: 4x – x = 3x. Then combine the y terms: 3y + 2y = 5y. The result is 3x + 5y.

例：4x + 3y – x + 2y。先合并含 x 的项：4x – x = 3x。再合并含 y 的项：3y + 2y = 5y。结果是 3x + 5y。

Remember that the order of terms in the final answer does not matter, but it is common to write letters in alphabetical order.

记住，最终答案中各项的顺序并不重要，但通常按字母顺序书写。

6. Dealing with Powers (Exponents) | 处理幂（指数）

When collecting like terms, the variable must match exactly, including its exponent. x and x² are not like terms.

合并同类项时，变量必须完全一致，包括它的指数。x 和 x² 不是同类项。

Example: 2x² + 3x² = 5x². The exponent 2 is unchanged because we are adding the coefficients only.

例：2x² + 3x² = 5x²。指数 2 不变，因为我们只合并系数。

However, 2x² + 3x cannot be simplified further because one term is quadratic in x and the other is linear in x.

然而，2x² + 3x 不能继续化简，因为一项是 x 的二次项，另一项是 x 的一次项。

In an expression like 5x³ – 2x + 4x³, combine only the cubic terms:

在类似 5x³ – 2x + 4x³ 的表达式中，只合并三次项：

5x³ + 4x³ – 2x = 9x³ – 2x

The -2x term stays separate because it has exponent 1.

-2x 项保持独立，因为它的指数是 1。

For products such as x²y and 3xy², note that they are not like terms because the exponents of x and y are different.

对于 x²y 和 3xy² 这样的乘积，注意它们不是同类项，因为 x 和 y 的指数不同。

7. Using the Distributive Property to Reveal Like Terms | 用分配率揭示同类项

Sometimes an expression contains brackets. Expanding the brackets with the distributive property can reveal like terms that were hidden.

有时表达式含有括号。用乘法分配律展开括号后，可以暴露出隐藏的同类项。

The distributive property states that a(b + c) = ab + ac.

乘法分配律规定：a(b + c) = ab + ac。

Example 1: Simplify 2(3x + 4) + 5x.

例1：化简 2(3x + 4) + 5x。

First expand: 2(3x + 4) = 6x + 8. Then add 5x: 6x + 8 + 5x = (6x + 5x) + 8 = 11x + 8.

先展开：2(3x + 4) = 6x + 8。然后加上 5x：6x + 8 + 5x = (6x + 5x) + 8 = 11x + 8。

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

例2：化简 3(2x – 1) – 2(4x + 3)。

Expand both brackets: 3(2x – 1) = 6x – 3 and -2(4x + 3) = -8x – 6. Put them together:

展开两个括号：3(2x – 1) = 6x – 3，-2(4x + 3) = -8x – 6。合在一起：

6x – 3 – 8x – 6 = (6x – 8x) + (-3 – 6) = -2x – 9

Always expand carefully, especially when subtracting a bracket, because the negative sign applies to every term inside.

展开时要小心，尤其是减去一个括号时，因为负号会作用于括号内的每一个项。

8. Simplifying Expressions with Brackets | 带括号的表达式化简

When an expression contains nested brackets, expand the innermost bracket first and work outward.

当表达式含有嵌套括号时，先从最内层的括号展开，再逐层向外。

Example: 4x – [2(3x – 1) + x]. Start with the inner bracket: 2(3x – 1) = 6x – 2.

例：4x – [2(3x – 1) + x]。先处理内层括号：2(3x – 1) = 6x – 2。

Then the expression becomes 4x – [6x – 2 + x] = 4x – [7x – 2].

然后表达式变为 4x – [6x – 2 + x] = 4x – [7x – 2]。

Now remove the outer bracket by subtracting each term: 4x – 7x + 2 = -3x + 2.

现在去掉外层括号，逐项相减：4x – 7x + 2 = -3x + 2。

It is helpful to write down each step on a new line so that you can check your work.

建议每一步写在新的一行，以便检查你的计算过程。

9. Real-World Applications: Perimeter Problems | 实际应用：周长问题

Combining like terms is often used to form simplified expressions for lengths, perimeters, and areas.

合并同类项常用于化简表示长度、周长和面积的表达式。

A rectangle has length 3x + 2 and width 2x – 1. Its perimeter is twice the length plus twice the width:

一个长方形的长是 3x + 2，宽是 2x – 1。它的周长是两倍长加两倍宽：

P = 2(3x + 2) + 2(2x – 1) = 6x + 4 + 4x – 2 = 10x + 2

So the perimeter is 10x + 2 units.

所以周长是 10x + 2 个单位。

Similarly, if a triangle has side lengths 2a, 3a + 5, and a – 2, its perimeter is:

类似地，如果三角形的三边长为 2a、3a + 5 和 a – 2，则其周长为：

2a + (3a + 5) + (a – 2) = 2a + 3a + 5 + a – 2 = 6a + 3

This simplified expression is easier to use if the value of a changes later.

这个化简后的表达式在之后 a 的数值发生变化时更容易使用。

10. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Mistake 1: Combining terms with different variables, such as 3x + 2y = 5xy. This is incorrect because xy is not the sum of x and y.

错误1：合并不同变量的项，例如 3x + 2y = 5xy。这是错误的，因为 xy 不是 x 与 y 的和。

Mistake 2: Adding exponents when combining terms. For example, x + x² is not x³. The terms are not like terms.

错误2：合并项时把指数相加。例如，x + x² 不是 x³。这两个项不是同类项。

Mistake 3: Forgetting the coefficient 1. The term x has coefficient 1, so x + 2x = 3x, not 2x².

错误3：忘记系数 1。项 x 的系数是 1，所以 x + 2x = 3x，而不是 2x²。

Mistake 4: Sign errors when subtracting brackets. Always multiply every term inside the bracket by the number before it, including its sign.

错误4：减去括号时出现符号错误。始终将括号前的数乘以括号内的每一项，同时注意符号。

To avoid these mistakes, first circle or highlight each distinct group of like terms before combining them.

要避免这些错误，可以先圈出或标出每一组同类项，再进行合并。

11. Practice Questions | 练习

Try the following questions. Simplify each expression fully.

尝试以下练习。将每个表达式完全化简。

Simplify 7a + 2a – 5a. | 化简 7a + 2a – 5a。
Simplify 3x + 4y – x + 2y. | 化简 3x + 4y – x + 2y。
Simplify 5m² + 3m² – 2m. | 化简 5m² + 3m² – 2m。
Simplify 4(2x + 1) – 3(x – 2). | 化简 4(2x + 1) – 3(x – 2)。
Simplify 2x(3x – 4) + 5x² – 2x. | 化简 2x(3x – 4) + 5x² – 2x。

Answers: 4a; 2x + 6y; 8m² – 2m; 5x + 10; 11x² – 10x.

答案：4a；2x + 6y；8m² – 2m；5x + 10；11x² – 10x。

Review each step to ensure you understand how the terms were combined.

请回顾每一步，确保你理解各项是如何合并的。

12. Summary | 总结

Collecting like terms means adding or subtracting the coefficients of terms that have the same variable factors and the same exponents.

合并同类项是指对具有相同变量因式和相同指数的项，将其系数相加或相减。

The variable part never changes when you combine like terms. Only the numerical coefficient is affected.

合并同类项时，变量部分永远不变，只有数字系数发生变化。

Different variables, or the same variable with different powers, cannot be combined into a single term.

不同的变量，或相同变量的不同幂，不能合并为一个单独的项。

With practice, collecting like terms becomes automatic. It is a vital skill for simplifying expressions, solving equations, and working with algebraic fractions.

通过练习，合并同类项会变得得心应手。这是化简表达式、解方程和处理代数分式的重要技能。

Always write down your working clearly, check the signs, and simplify until no more like terms remain.

始终清楚地写出解题过程，检查符号，并化简到不再有同类项为止。

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Collecting Like Terms | 合并同类项