📚 Collecting Like Terms | 合并同类项
In algebra, expressions often contain several terms that can be simplified. Collecting like terms is the process of combining terms that have exactly the same variable part and power. This skill is essential for IGCSE algebra, equation solving and many later topics.
在代数中，表达式常常包含多个可以化简的项。合并同类项就是把含有完全相同变量部分和幂次的项组合在一起的过程。这项技能对 IGCSE 代数、解方程以及后续许多主题都至关重要。
1. What Are Like Terms? | 什么是同类项
Like terms are terms whose variable parts are identical. The numbers in front, called coefficients, can be different, but the letters and their powers must match exactly.
同类项是指变量部分完全相同的项。前面的数字叫做系数，系数可以不同，但字母及其幂次必须完全一致。
For example, 3x and -5x are like terms because both contain the variable x to the power 1. The terms 4x² and 4x are not like terms because the powers of x are different.
例如，3x 和 -5x 是同类项，因为两者都含有 x 的一次方。4x² 和 4x 不是同类项，因为 x 的幂次不同。
| Pair of terms | Like terms? | Reason |
|---|---|---|
| 3x and -5x | Yes | Same variable x, same power 1 |
| 2x² and -x² | Yes | Same variable x, same power 2 |
| 5xy and 3xy | Yes | Same variables x and y |
| 4x and 4x² | No | Powers of x are different |
| 6a and 6b | No | Different variables |
Constant terms such as 4 and -7 are also like terms because they have no variable part at all.
常数项如 4 和 -7 也是同类项，因为它们完全没有变量部分。
2. Why Collect Like Terms? | 为什么合并同类项
Collecting like terms makes an expression shorter and easier to understand. A simplified expression is also easier to substitute into, factorise or use in solving equations.
合并同类项可以使表达式更短、更易理解。化简后的表达式在代入、因式分解或解方程时也更加方便。
For example, the expression 2x + 3y + 5x - y contains four terms. After collecting like terms, it becomes 7x + 2y, which is much simpler.
例如，表达式 2x + 3y + 5x - y 包含四个项。合并同类项后，它变为 7x + 2y，简单得多。
In IGCSE questions, you will often need to simplify before solving, rearranging or proving an identity. Collecting like terms is usually the first step.
在 IGCSE 题目中，你常常需要先化简，再解方程、移项或证明恒等式。合并同类项通常是第一步。
3. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and look only at the variable part. Check both the letter and its power.
要识别同类项，先忽略系数，只看变量部分。同时检查字母及其幂次。
For example, 7xy and -2xy are like terms, but 7xy and 7x are not, because the first pair has both x and y while the second pair has only x.
例如，7xy 和 -2xy 是同类项，但 7xy 和 7x 不是，因为第一对含有 x 和 y 两个变量，而第二对只有 x。
Be careful with powers: x, x² and x³ are three different variable parts, so terms containing them cannot be combined.
注意幂次：x、x² 和 x³ 是三种不同的变量部分，因此含有它们的项不能合并。
- 4a and 9a → like terms
- 4a and 9a² → not like terms
- -3mn and 8mn → like terms
- -3mn and 8m → not like terms
上述列表展示了如何通过字母和幂次判断同类项。
4. Adding and Subtracting Like Terms | 同类项的加减法
To add or subtract like terms, add or subtract their coefficients and keep the variable part unchanged.
要加减同类项，只需加减它们的系数，变量部分保持不变。
3x + 5x = 8x
In this example, the coefficients 3 and 5 are added to give 8, and the x is kept exactly the same.
在这个例子中，系数 3 和 5 相加得到 8，x 保持完全不变。
7y - 2y = 5y
Here the coefficients are subtracted: 7 - 2 = 5, so the result is 5y.
这里系数相减：7 - 2 = 5，因此结果是 5y。
For longer expressions, first group the like terms together, then combine each group separately.
对于较长的表达式，先把同类项分组，再分别合并每一组。
2x + 3y + 5x - y = 2x + 5x + 3y - y = 7x + 2y
5. Working with Negative Coefficients | 处理负系数
Negative coefficients cause many sign errors. Use the sign in front of each term as part of its coefficient.
负系数常常导致符号错误。要把每一项前面的符号看作其系数的一部分。
4x - 9x = -5x
Since 4 - 9 = -5, the result is -5x. The variable part does not disappear.
因为 4 - 9 = -5，结果是 -5x。变量部分不会消失。
-2a - 3a = -5a
Here both coefficients are negative, so they are added in magnitude and the result keeps the negative sign.
这里两个系数都是负数，所以绝对值相加，结果保留负号。
For mixed expressions, combine positive and negative coefficients carefully.
对于混合表达式，要小心地合并正系数和负系数。
6p - 2p + 3p = 7p
6p - 10p + 3p = -1p = -p
In the second line, 6 - 10 + 3 = -1, so the simplified term is -p, not p.
在第二行中，6 - 10 + 3 = -1，所以化简后的项是 -p，而不是 p。
6. Expanding Brackets Before Collecting | 先去括号再合并
When an expression contains brackets, you must expand them first before collecting like terms.
当表达式含有括号时，必须先展开括号，然后再合并同类项。
3(x + 2) + 2(x - 1)
First distribute the 3 and the 2:
首先分别将 3 和 2 乘入括号：
3x + 6 + 2x - 2
Now collect the x terms and the constant terms:
现在合并 x 项和常数项：
3x + 2x + 6 - 2 = 5x + 4
Remember that a minus sign before a bracket changes the sign of every term inside the bracket.
记住，括号前的负号会改变括号内每一项的符号。
5 - 2(x - 3) = 5 - 2x + 6 = 11 - 2x
7. Simplifying Expressions with Powers | 含幂项的化简
Terms with different powers of the same letter are not like terms and cannot be combined.
同一个字母的不同幂次不是同类项，不能合并。
2x² + 3x
This expression is already simplified because x² and x are not like terms.
这个表达式已经是最简形式，因为 x² 和 x 不是同类项。
However, terms with the same power can be combined.
但是，相同幂次的项可以合并。
4x² - x² = 3x²
Also note that 1x² is usually written as x², and -1x² is written as -x².
还要注意，1x² 通常写作 x²，-1x² 写作 -x²。
For mixed powers, keep each power group separate:
对于混合幂次，每一组幂次分开保留：
2x² + 5x - 3x² + 2x = -x² + 7x
8. Common Mistakes | 常见错误
One common mistake is adding unlike terms, such as writing 3x + 4x² as 7x². This is incorrect because x and x² are different variables parts.
一个常见错误是把不同类项相加，例如把 3x + 4x² 写成 7x²。这是错误的，因为 x 和 x² 是不同的变量部分。
Another mistake is changing the variable part when adding coefficients. For example, 5a + 2a should be 7a, not 7a².
另一个错误是在相加系数时改动了变量部分。例如，5a + 2a 应该是 7a，而不是 7a²。
Sign errors are also common. When subtracting a term, remember to keep the negative sign with the coefficient.
符号错误也很常见。减去某一项时，记住让负号跟随着系数。
Finally, forgetting to expand brackets first will give an incorrect answer. Always expand before you collect.
最后，忘记先展开括号会得到错误答案。一定要先去括号再合并。
9. Worked Examples | 例题解析
Example 1: Simplify 7x + 3 - 2x + 8.
例题 1：化简 7x + 3 - 2x + 8。
Group like terms: 7x - 2x + 3 + 8 = 5x + 11.
合并同类项：7x - 2x + 3 + 8 = 5x + 11。
Example 2: Simplify 4(2a - 1) - 3(a + 2).
例题 2：化简 4(2a - 1) - 3(a + 2)。
Expand: 8a - 4 - 3a - 6 = 5a - 10.
展开：8a - 4 - 3a - 6 = 5a - 10。
Example 3: Simplify 2x² + 3x - x² + 4x - 5.
例题 3：化简 2x² + 3x - x² + 4x - 5。
Collect like terms: 2x² - x² + 3x + 4x - 5 = x² + 7x - 5.
合并同类项：2x² - x² + 3x + 4x - 5 = x² + 7x - 5。
10. Quick Checklist | 快速检查清单
Before you finish an IGCSE simplifying question, check these points:
在完成 IGCSE 化简题之前，请检查以下几点：
- All brackets have been expanded first.
- Only like terms have been combined.
- Variable parts have not been changed.
- Negative signs have been handled correctly.
- The final expression has no further like terms.
先展开所有括号；只合并同类项；不要改变变量部分；正确处理负号；最终表达式不再含有可合并的同类项。
Collecting like terms is a foundational algebra skill. Practice it regularly so that it becomes automatic in every IGCSE algebra question.
合并同类项是一项基础代数技能。定期练习，使它在每道 IGCSE 代数题中都能自动完成。
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