Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In algebra, simplifying expressions is a fundamental skill. Combining like terms is the process of grouping and adding or subtracting terms that have the same variable parts. This makes expressions shorter, cleaner, and easier to evaluate.

在代数学中，化简表达式是一项基本技能。合并同类项是将具有相同变量部分的项进行分组并相加或相减的过程。它使表达式更简短、更清晰，也更容易求值。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variables raised to the same powers. Only the numerical coefficients may be different. For example, 3x and 5x are like terms because both contain the variable x to the first power.

同类项是指具有完全相同变量且变量对应指数也相同的项。只有数字系数可以不同。例如，3x 和 5x 是同类项，因为它们都包含一次方的变量 x。

Constant terms are also like terms. The numbers 4 and -7 are like terms because they have no variable part. But 2x and 2x² are not like terms, because the powers of x are different.

常数项也是同类项。数字 4 和 -7 是同类项，因为它们没有变量部分。但 2x 和 2x² 不是同类项，因为 x 的指数不同。

3x and 9x → like terms
4y² and -2y² → like terms
5x and 5x² → not like terms
7 and 7x → not like terms

3x 和 9x → 同类项
4y² 和 -2y² → 同类项
5x 和 5x² → 不是同类项
7 和 7x → 不是同类项

2. Identifying Like Terms | 识别同类项

To identify like terms, look at the variable part of each term. Ignore the coefficient (the number in front). The variable letters and their exponents must match exactly.

要识别同类项，请观察每一项的变量部分。忽略系数（前面的数字）。变量字母及其指数必须完全匹配。

For example, in the expression 4a + 3b – 2a + 5b, the terms 4a and -2a are like terms, while 3b and 5b are another pair of like terms.

例如，在表达式 4a + 3b – 2a + 5b 中，4a 和 -2a 是同类项，而 3b 和 5b 是另一组同类项。

4a + 3b – 2a + 5b = (4a – 2a) + (3b + 5b)

3. Adding and Subtracting Like Terms | 同类项的加法和减法

When combining like terms, add or subtract their coefficients. The variable part stays exactly the same.

合并同类项时，对其系数进行加或减，变量部分保持不变。

For example, 6x + 2x = 8x. Here, 6 + 2 = 8, and x remains unchanged. Similarly, 9y – 4y = 5y.

例如，6x + 2x = 8x。这里，6 + 2 = 8，x 保持不变。同样，9y – 4y = 5y。

6x + 2x = 8x, 9y – 4y = 5y

6x + 2x = 8x， 9y – 4y = 5y

If a term has no written coefficient, the coefficient is 1. For example, x means 1x, so x + 3x = 4x.

如果一项没有写出系数，则系数为 1。例如，x 表示 1x，所以 x + 3x = 4x。

4. Combining Terms with Different Signs | 合并带不同符号的项

Always include the sign in front of a term when combining. A negative sign means that term is subtracted.

合并时始终包含项前面的符号。负号表示该项被减去。

For example, in 5x – 3x + 2x, combine 5 – 3 + 2 = 4, so the result is 4x.

例如，在 5x – 3x + 2x 中，合并 5 – 3 + 2 = 4，因此结果是 4x。

Be careful with expressions like -7a + 3a. The coefficients are -7 and 3. Their sum is -4, so -7a + 3a = -4a.

注意像 -7a + 3a 这样的表达式。系数是 -7 和 3。它们的和是 -4，所以 -7a + 3a = -4a。

-7a + 3a = -4a

5. Simplifying Expressions with Multiple Variables | 含多个变量的表达式化简

When an expression contains more than one variable, group each type of like term separately.

当表达式包含多个变量时，分别对每种同类项进行分组。

Simplify 2x + 3y – x + 4y + 5. First, group x terms: 2x – x = x. Then y terms: 3y + 4y = 7y. The constant is 5. So the simplified expression is x + 7y + 5.

化简 2x + 3y – x + 4y + 5。首先，合并 x 项：2x – x = x。然后 y 项：3y + 4y = 7y。常数项是 5。所以化简后的表达式是 x + 7y + 5。

You can rearrange terms to put them together, but keep the sign that belongs to each term.

可以重新排列项使它们放在一起，但要保留每项所属的符号。

6. Using the Distributive Property | 运用分配律

The distributive property allows you to remove brackets before combining like terms. The rule is a(b + c) = ab + ac.

分配律允许你在合并同类项之前去掉括号。规则是 a(b + c) = ab + ac。

Simplify 3(x + 2) + 4x. First expand: 3x + 6 + 4x. Then combine x terms: 3x + 4x = 7x. The result is 7x + 6.

化简 3(x + 2) + 4x。首先展开：3x + 6 + 4x。然后合并 x 项：3x + 4x = 7x。结果是 7x + 6。

3(x + 2) + 4x = 3x + 6 + 4x = 7x + 6

7. Combining Like Terms in Brackets | 括号中的合并

Sometimes brackets contain like terms that can be simplified before expanding. For example, (5x + 3x) can be combined first: 8x.

有时括号内包含的同类项可以先合并再展开。例如，(5x + 3x) 可以先合并：8x。

More often, you need to expand brackets first. Consider 2(x + 4) – (x – 1). Expand: 2x + 8 – x + 1. Note the negative in front of the second bracket: it changes both signs. Now combine: 2x – x = x, and 8 + 1 = 9. The answer is x + 9.

更常见的是需要先展开括号。考虑 2(x + 4) – (x – 1)。展开：2x + 8 – x + 1。注意第二个括号前的负号，它会改变括号内的两个符号。现在合并：2x – x = x，8 + 1 = 9。答案是 x + 9。

2(x + 4) – (x – 1) = 2x + 8 – x + 1 = x + 9

8. Common Mistakes | 常见错误

Students often make mistakes when combining terms that are not like terms. Remember that 2x and 2y cannot be combined, and x and x² cannot be combined.

学生在合并非同类项时经常出错。记住 2x 和 2y 不能合并，x 和 x² 也不能合并。

Wrong: 3x + 5y = 8xy
Correct: 3x + 5y cannot be simplified

错误： 3x + 5y = 8xy
正确： 3x + 5y 不能化简

Another common mistake is forgetting the sign in front of a term. When moving terms, keep their signs attached.

另一个常见错误是忘记项前面的符号。移动项时，要连同其符号一起移动。

9. Practice Problems | 练习题

Try simplifying these expressions on your own.

请尝试自己化简以下表达式。

Problem 1	7x + 2x
Problem 2	5a – 3a + 2
Problem 3	4x + 3y – 2x + y
Problem 4	2(x + 3) + 4x
Problem 5	3x² + 2x – x² + 5x

练习 1	7x + 2x
练习 2	5a – 3a + 2
练习 3	4x + 3y – 2x + y
练习 4	2(x + 3) + 4x
练习 5	3x² + 2x – x² + 5x

10. Solutions and Explanations | 答案与讲解

Check your answers and see the step-by-step reasoning below.

核对你的答案，并查看下面的逐步推理。

Problem 1: 7x + 2x = (7 + 2)x = 9x.

练习 1： 7x + 2x = (7 + 2)x = 9x。

Problem 2: 5a – 3a + 2 = (5 – 3)a + 2 = 2a + 2.

练习 2： 5a – 3a + 2 = (5 – 3)a + 2 = 2a + 2。

Problem 3: 4x + 3y – 2x + y = (4x – 2x) + (3y + y) = 2x + 4y.

练习 3： 4x + 3y – 2x + y = (4x – 2x) + (3y + y) = 2x + 4y。

Problem 4: 2(x + 3) + 4x = 2x + 6 + 4x = 6x + 6.

练习 4： 2(x + 3) + 4x = 2x + 6 + 4x = 6x + 6。

Problem 5: 3x² + 2x – x² + 5x = (3x² – x²) + (2x + 5x) = 2x² + 7x.

练习 5： 3x² + 2x – x² + 5x = (3x² – x²) + (2x + 5x) = 2x² + 7x。

11. Real-World Applications | 实际应用

Combining like terms appears in many real-life situations, such as calculating total costs or distances. If one item costs 3a dollars and another item costs 5a dollars, the total cost is 8a dollars.

合并同类项出现在许多现实场景中，例如计算总成本或总距离。如果一件物品花费 3a 美元，另一件物品花费 5a 美元，则总成本为 8a 美元。

In geometry, the perimeter of a rectangle with length 2x and width x can be written as 2(2x) + 2(x) = 4x + 2x = 6x. This simplified expression is easier to use.

在几何中，长为 2x、宽为 x 的矩形周长可以写成 2(2x) + 2(x) = 4x + 2x = 6x。这个化简后的表达式更易使用。

Physics and engineering also use algebraic simplification to reduce complex formulas before solving problems.

物理和工程学也使用代数化简来在解题前减少复杂公式。

12. Conclusion | 总结

Combining like terms is a core algebraic technique. Always check that the variables and exponents match, keep the signs intact, and use the distributive property when expanding brackets.

合并同类项是代数的核心技巧。始终检查变量和指数是否匹配，保持符号不变，并在展开括号时使用分配律。

With regular practice, simplifying expressions becomes quick and automatic. This skill is essential for solving equations, working with functions, and mastering higher-level mathematics.

通过经常练习，化简表达式会变得快速而自然。这项技能对于解方程、处理函数以及掌握更高阶数学至关重要。

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Combining Like Terms | 合并同类项