📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, collecting like terms is the first essential step in simplifying expressions. It allows you to tidy up an expression so that equations become easier to solve, brackets become easier to expand, and answers can be written in their simplest form. This skill is tested constantly across algebra topics.
在 IGCSE 代数中，合并同类项是化简表达式的首要基本步骤。它能帮助你整理表达式，使方程更易求解、括号更易展开、答案可以写成最简形式。这项技能在代数各类题目中都会反复考查。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable part, including the same letters and the same powers. In other words, only the numbers in front of the variables may be different. For example, 3x and 5x are like terms because both contain the variable x to the power 1.
同类项是指含有完全相同字母部分且每个字母指数也相同的项。换句话说，只有变量前面的数字可以不同。例如，3x 和 5x 是同类项，因为两者都含有一次方的变量 x。
3x + 2x = 5x
The terms 4x² and 7x² are also like terms because both contain x². However, 4x and 4x² are not like terms, because the powers of x are different.
4x² 和 7x² 也是同类项，因为两者都含有 x²。但是 4x 和 4x² 不是同类项，因为 x 的指数不同。
- 3a and 9a are like terms.
- 3a and 9a 是同类项。
- 5xy and −2xy are like terms.
- 5xy 和 −2xy 是同类项。
- 6m and 6n are not like terms.
- 6m 和 6n 不是同类项。
2. Key Rules for Collecting Like Terms | 合并同类项的核心规则
variable as the object you are counting: 3 apples + 2 apples = 5 apples, so 3x + 2x = 5x.
合并同类项时，字母部分完全保持不变，只对系数进行加减。可以把变量想象成你在计数的对象：3 个苹果加 2 个苹果等于 5 个苹果，所以 3x + 2x = 5x。
7a − 2a = (7 − 2)a = 5a
+ x = 2x, not x². The exponent stays as 1 unless the question asks you to multiply powers.
千万不要对指数进行加减。例如，x + x = 2x，而不是 x²。除非题目要求进行幂的乘法运算，否则指数保持不变。
3. Positive and Negative Coefficients | 正系数与负系数
Each term carries the sign in front of it. In 4x − 3x + 2x, the second term is −3x, so its coefficient is −3, not 3. This is important because losing a negative sign is a very common mistake.
每一项都带有它前面的符号。在 4x − 3x + 2x 中，第二项是 −3x，所以它的系数是 −3，而不是 3。这一点很重要，因为丢失负号是非常常见的错误。
4x − 3x + 2x = (4 − 3 + 2)x = 3x
When terms are rearranged, move the sign together with the term. For example, 5 − 2x + 3x can be written as 5 + 3x − 2x, then simplified to 5 + x.
重新排列各项时，符号要和对应项一起移动。例如，5 − 2x + 3x 可以写成 5 + 3x − 2x，然后化简为 5 + x。
4. Different Powers Are Not Like Terms | 不同次幂不是同类项
Terms with different powers of the same variable cannot be combined. For instance, x, x² and x³ are different kinds of terms. You can only collect terms that have exactly the same power.
同一字母的不同次幂不能合并。例如，x、x² 和 x³ 是不同类型的项。你只能合并次幂完全相同的项。
x² + 2x² = 3x²
x² + x cannot be combined further
In an expression like 2x² + 3x − x² + 5x, group the x² terms and the x terms separately. The result is x² + 8x.
在 2x² + 3x − x² + 5x 这样的表达式中，要分别合并 x² 项和 x 项。结果是 x² + 8x。
5. Worked Example 1: Simple Collection | 例题1：简单合并
like terms because they all contain y to the power 1. Add the coefficients: 5 + 3 − 1 = 7.
化简 5y + 3y − y。这三项都是同类项，因为它们都含有一次方的 y。将系数相加：5 + 3 − 1 = 7。
5y + 3y − y = 7y
Be careful with the term −y. It has an invisible coefficient of −1. Writing it as −1y can help you avoid mistakes in the calculation.
要注意 −y 这一项。它有一个隐形的系数 −1。把它写成 −1y 有助于你在计算时避免出错。
6. Worked Example 2: With Negative Terms | 例题2：含负项
Simplify 6x − 2 − 3x + 5. First, group the x terms and the constant terms: 6x − 3x gives 3x, and −2 + 5 gives 3.
化简 6x − 2 − 3x + 5。首先把 x 项和常数项分别分组：6x − 3x 得到 3x，−2 + 5 得到 3。
6x − 2 − 3x + 5 = 3x + 3
The final answer is 3x + 3. The two parts cannot be combined further because one contains the variable x and the other is a constant.
最终答案是 3x + 3。这两部分不能再进一步合并，因为一个含有变量 x，另一个是常数。
7. Combining More Than Two Sets | 合并两组以上同类项
Many expressions contain more than one type of like term. Simplify 4a + 3b − 2a + 5b by collecting the a terms and b terms separately.
许多表达式包含不止一组同类项。化简 4a + 3b − 2a + 5b 时，要分别合并 a 项和 b 项。
4a − 2a = 2a
3b + 5b = 8b
4a + 3b − 2a + 5b = 2a + 8b
Always check that your final expression has no remaining like terms. In this case, 2a and 8b are not like terms, so the expression is fully simplified.
始终检查最终表达式是否还有可合并的同类项。这里 2a 和 8b 不是同类项，因此表达式已经彻底化简。
8. Fractions and Decimals | 分数与小数
The same rules apply when coefficients are fractions or decimals. For fractions, use a common denominator before adding. For decimals, add the numbers carefully by place value.
当系数是分数或小数时，合并规则完全相同。对于分数，先通分再进行加法；对于小数，按数位仔细相加。
½x + ⅓x = ⅚x
Here the common denominator of 2 and 3 is 6, so ½ becomes 3/6 and ⅓ becomes 2/6, giving a total of 5/6.
这里 2 和 3 的公分母是 6，所以 ½ 变为 3/6，⅓ 变为 2/6，总和为 5/6。
is often a good idea to convert decimals to fractions if the final answer needs to be exact.
对于小数，0.5x + 0.25x = 0.75x。如果最终答案需要精确值，通常把小数化为分数会更好。
9. Using Substitution to Check | 代入检验
e, if the original is 6x − 2 − 3x + 5 and the simplified form is 3x + 3, substitute x = 2.
你可以用代入法检验化简结果：把同一个数值代入原式和化简后的表达式，两边结果必须相等。例如，若原式是 6x − 2 − 3x + 5，化简形式为 3x + 3，代入 x = 2。
Original: 6(2) − 2 − 3(2) + 5 = 12 − 2 − 6 + 5 = 9
Simplified: 3(2) + 3 = 6 + 3 = 9
Since both give 9, the simplification is correct. Substitution is a fast way to catch sign errors or missed terms in an exam.
因为两者都得到 9，说明化简正确。在考试中，代入法是一种快速发现符号错误或漏项的方法。
10. Common Mistakes | 常见错误
cause you are counting how many x terms there are, not multiplying the variable by itself.
最常见的错误之一是合并不同类项，例如把 x + x 写成 x²。正确结果是 2x，因为你是在计算有多少个 x 项，而不是把变量自身相乘。
- Losing a negative sign: 3x − 5x = −2x, not 2x.
- 丢失负号：3x − 5x = −2x，而不是 2x。
- Changing the power: 2x² + 3x² = 5x², not 5x⁴.
- 改变指数：2x² + 3x² = 5x²，而不是 5x⁴。
- Forgetting the invisible 1: x + 2x = 3x, not 2x.
- 忘记隐形系数 1：x + 2x = 3x，而不是 2x。
11. Exam-Style Questions | 考试题型
IGCSE questions often ask you to simplify an expression before substituting into a formula or solving an equation. Practice by simplifying these expressions fully.
IGCSE 考题经常要求你先化简表达式，再代入公式或求解方程。请练习完全化简以下表达式。
Simplify 7p − 4q + 2p + 9q
The p terms give 7p + 2p = 9p. The q terms give −4q + 9q = 5q. So the answer is 9p + 5q.
p 项相加得到 7p + 2p = 9p。q 项相加得到 −4q + 9q = 5q。所以答案是 9p + 5q。
Simplify 8m² − 3m + 4 − m² + 6m − 10
Collecting the m² terms gives 7m². Collecting the m terms gives 3m. Collecting the constants gives −6. The final result is 7m² + 3m − 6.
合并 m² 项得到 7m²，合并 m 项得到 3m，合并常数项得到 −6。最终结果是 7m² + 3m − 6。
12. Summary Checklist | 复习清单
By now you should be able to identify like terms, collect them correctly, and avoid the common errors that lose marks in exams. Use this checklist before moving on to harder algebra topics.
现在你应该能够识别同类项、正确合并它们，并避免考试中常见的失分错误。在进入更难的代数主题之前，请对照这份清单进行检查。
- I can identify terms with the same variable and power.
- 我能识别字母和指数都相同的项。
- I can add and subtract coefficients without changing the variable part.
- 我能只对系数进行加减而不改变字母部分。
- I can handle negative signs correctly.
- 我能正确处理负号。
- I can simplify expressions with fractions and decimals.
- 我能化简含有分数和小数的表达式。
- I can use substitution to check my answer.
- 我能用代入法检查答案。
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