📚 Combining Like Terms | 合并同类项
Combining like terms, also called collecting like terms, is one of the most important skills in algebra. It allows us to write an expression in a simpler form without changing its value.
合并同类项是代数学习中最重要、最基础的技能之一。它让我们在不改变原式数值的前提下，把一个代数表达式写成更简单的形式。
1. What Are Algebraic Terms? | 什么是代数项？
An algebraic term is a number multiplied by one or more variables. For example, 3x, -2y, 5a²b and -7 are all terms.
代数项是数与一个或多个变量相乘组成的整体。例如：3x、-2y、5a²b 和 -7 都是代数项。
- The number part is called the coefficient. 数乘部分称为系数。
- The letter part is called the variable part. 字母部分称为变量部分。
In the term 3x, the coefficient is 3 and the variable part is x. In -2y, the coefficient is -2 and the variable part is y.
在项 3x 中，系数是 3，变量部分是 x；在 -2y 中，系数是 -2，变量部分是 y。
2. Like Terms vs Unlike Terms | 同类项与异类项
Like terms have exactly the same variable part, with the same powers. Unlike terms have different variables or different powers.
同类项是指所含字母完全相同，并且相同字母的指数也完全相同的项；异类项则字母不同或指数不同。
Like terms: 3x and 7x; 5x²y and -2x²y
Unlike terms: 3x and 3y; 4x and 4x²
| Terms / 项 | Like? / 是否同类 | Reason / 理由 |
| 2x and 5x | Yes / 是 | Same variable x, same power 1 / 字母同为 x，指数同为 1 |
| 2x and 2y | No / 否 | Different variables / 字母不同 |
| 3a² and 3a | No / 否 | Powers of a are different / a 的指数不同 |
| 4ab and -7ab | Yes / 是 | Same variables a and b / 字母 a 和 b 相同 |
Constant numbers are also like terms. For example, 6 and -10 are like terms because neither contains a variable.
常数项也是同类项。例如，6 和 -10 都是常数项，它们都只包含数字，不包含字母。
3. The Rule for Combining Like Terms | 合并同类项的规则
To combine like terms, keep the variable part exactly the same and add or subtract the coefficients.
合并同类项时，保持变量部分完全相同，只把各项的系数相加或相减。
ax + bx = (a + b)x
For example, 3x + 5x means "3 copies of x" plus "5 copies of x". That gives 8 copies of x, so 3x + 5x = 8x.
例如，3x + 5x 表示“3 个 x”加上“5 个 x”，一共得到“8 个 x”，所以 3x + 5x = 8x。
This rule works because of the distributive property: (3 + 5)x = 8x.
这个规则成立是因为乘法分配律：(3 + 5)x = 8x。
4. Combining Positive Coefficients | 合并正系数
When all coefficients are positive, simply add them.
当所有系数都是正数时，直接相加即可。
2x + 3x = 5x
4y + y = 5y
Notice that y means 1y, so 4y + y = 4y + 1y = 5y.
注意 y 就是 1y，因此 4y + y = 4y + 1y = 5y。
This idea is like adding fruits: 2 apples plus 3 apples is 5 apples. Here x stands for the object we are counting.
这个思路就像把水果相加：2 个苹果加 3 个苹果等于 5 个苹果。这里的 x 表示我们在计数的对象。
5. Combining Negative Coefficients | 合并负系数
When some coefficients are negative, follow the sign of each term carefully.
当有的系数是负数时，要特别注意每一项前面的正负号。
5a - 7a + 2a = (5 - 7 + 2)a = 0a = 0
In this example, the result is zero, so the terms disappear from the expression.
在这个例子中，结果为 0，因此这些项在表达式中消失。
4x - 9x = -5x
Here 4 minus 9 gives -5, so the simplified term has a negative coefficient.
这里 4 减 9 得 -5，所以化简后的项系数为负数。
6. Combining x and x² Terms | 合并 x 与 x² 项
x and x² are not like terms. They must be kept separate in an expression.
x 与 x² 不是同类项，它们在表达式中必须分别处理。
2x + 3x² cannot be simplified
2x + 3x² 不能继续化简
Think of x as a length and x² as an area. A length plus an area does not have a simple single name.
可以把 x 看作长度，把 x² 看作面积。长度加面积不能直接合并成一个简单的同类项。
However, 2x + 3x is allowed, and 2x² + 3x² is allowed because they are like terms.
但是 2x + 3x 可以合并，2x² + 3x² 也可以合并，因为它们是同类项。
2x² + 3x² = 5x²
When combining x² terms, do not change the exponent. Only the coefficient is added.
合并 x² 项时，千万不要改变指数，只把系数相加。
7. Combining Constants | 合并常数项
Constant terms are numbers without variables. They are all like terms, so they can be combined.
常数项是不含字母的数字项。它们彼此都是同类项，因此可以合并。
7 - 2 = 5
-3 + 8 = 5
In expressions, constants are often written at the end of the simplified answer.
在表达式中，常数项通常写在化简结果的末尾。
For example, 4x + 6 - 2x + 5 = 2x + 11.
例如，4x + 6 - 2x + 5 = 2x + 11。
8. Combining Terms with Multiple Variables | 合并多变量项
Terms such as ab, ab², and a²b all have different variable parts and cannot be mixed.
ab、ab² 和 a²b 这些项的变量部分不同，不能混在一起合并。
3ab + 5ab = 8ab
4x²y - x²y = 3x²y
The variable part must match completely: x²y and xy² are not like terms because the powers on x and y are different.
变量部分必须完全相同：x²y 与 xy² 不是同类项，因为 x 和 y 的指数不同。
Think of ab as one object. If you have 3 ab's and 5 ab's, you have 8 ab's.
可以把 ab 看作一个整体。如果你有 3 个 ab 和 5 个 ab，那么一共就有 8 个 ab。
9. Stepping Through a Full Simplification | 完整化简示例
Let us simplify a longer expression step by step.
下面我们一步一步化简一个较长的代数表达式。
Example / 例：Simplify 5x + 3y - 2x + 4y + 7 - 2
Step 1: Identify and group the x terms: 5x - 2x = 3x.
第一步：找出并合并 x 项：5x - 2x = 3x。
Step 2: Group the y terms: 3y + 4y = 7y.
第二步：合并 y 项：3y + 4y = 7y。
Step 3: Combine the constants: 7 - 2 = 5.
第三步：合并常数项：7 - 2 = 5。
5x + 3y - 2x + 4y + 7 - 2 = 3x + 7y + 5
Notice that 3x and 7y are unlike terms, so they stay separate.
注意 3x 与 7y 是异类项，所以它们在结果中保持分开。
Here is another example with powers:
下面是一个含平方项的示例：
4x² + 3x - 2x² + 5x - 1 = 2x² + 8x - 1
After combining, it is often helpful to write terms in descending order of powers.
合并后，通常按字母指数的降序排列各项会更清晰。
10. Common Pitfalls | 常见易错点
- Do not combine 3x and 3y. They are unlike terms. 不要把 3x 和 3y 合并，它们是异类项。
- Do not combine 2x and 3x². They are not like terms. 不要把 2x 和 3x² 合并，它们不是同类项。
- Do not change the power when adding coefficients: 2x + 3x = 5x, not 5x². 合并系数时不要改变指数：2x + 3x = 5x，而不是 5x²。
- Keep the sign in front of each term. For example, 3x - 2x = x, not 5x. 注意保留每项前面的符号。例如 3x - 2x = x，不是 5x。
- Remember that x means 1x and -x means -1x. 记住 x 表示 1x，-x 表示 -1x。
11. Exam-Style Questions | 考题练习
Question 1 / 题目 1：Simplify 6p + 2q - 3p + 5q.
6p + 2q - 3p + 5q = (6p - 3p) + (2q + 5q) = 3p + 7q
Here p terms and q terms are combined separately.
这里 p 项与 q 项分别合并。
Question 2 / 题目 2：Simplify 4x² + 3x - 2x² - x.
4x² + 3x - 2x² - x = (4x² - 2x²) + (3x - x) = 2x² + 2x
Remember that -x is the same as -1x.
记住 -x 与 -1x 相同。
Question 3 / 题目 3：Simplify 8a - 4a + 3a.
8a - 4a + 3a = (8 - 4 + 3)a = 7a
Always show the extra working step if you are not confident mentally.
如果你对心算还不够自信，建议把中间步骤写出来。
12. Conclusion | 结语
Combining like terms is a core skill for all later algebra work, including solving equations, expanding brackets and factorising.
合并同类项是后续所有代数学习的基础，包括解方程、展开括号和因式分解。
Remember the simple rule: keep the variable part unchanged and add the coefficients. With this rule, any expression can be simplified clearly and accurately.
记住这条简单规则：保持变量部分不变，合并系数。掌握了它，你就能准确、清晰地化简各种代数表达式。
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