📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, collecting like terms is one of the first and most important algebra skills. It allows you to simplify expressions such as 3x + 5x - 2x into a single term, making later work with equations, brackets and graphs much easier.
在 IGCSE 数学中，合并同类项是最早也是最重要的代数技能之一。它能把 3x + 5x - 2x 这样的表达式化简为单项，让后续的方程、括号和图像问题简单很多。
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 powers. For example, 4x and 7x are like terms because both contain x to the power 1. The terms 2x² and 5x² are like terms because both contain x².
同类项是指字母部分完全相同（包括相同指数）的项。例如 4x 和 7x 是同类项，因为它们都含有 x 的一次方；2x² 和 5x² 是同类项，因为它们都含有 x²。
Constant terms such as 3, -8 and 12 are also like terms with each other, because they have no variable part at all.
常数项如 3、-8 和 12 也彼此是同类项，因为它们根本没有字母部分。
By contrast, 3x and 3x² are not like terms; one has x¹ and the other has x². Similarly, 3x and 3y are not like terms.
相比之下，3x 和 3x² 不是同类项；一个含 x¹，另一个含 x²。同样，3x 和 3y 也不是同类项。
2. Why 'Collecting' Matters | 为什么“合并”很重要
Algebraic expressions can look long and messy. Collecting like terms reduces them to a simpler form, which is essential for solving equations, sketching graphs and rearranging formulas.
代数表达式可能看起来又长又乱。合并同类项能把它们化简，这对于解方程、画图像和变形公式都很重要。
In an exam, a simplified answer is usually easier to check. It also reduces the chance of sign errors in later steps.
在考试中，化简后的答案通常更容易检查，也能减少后续步骤中的符号错误。
For example, 2a + 3b + 4a - b simplifies to 6a + 2b. This shorter form tells you immediately the total contribution of a and b.
例如，2a + 3b + 4a - b 化简为 6a + 2b。这个更短的形式能立即告诉你 a 与 b 的总贡献。
3. The Golden Rule: Only Coefficients Add | 黄金法则：只有系数相加
When collecting like terms, only the coefficients (the number parts) are added or subtracted. The variable part stays exactly the same.
合并同类项时，只有系数（数字部分）相加或相减，字母部分保持不变。
3x + 5x = (3 + 5)x = 8x
ecause multiplication distributes over addition: 3x means 3 × x, so 3 × x + 5 × x = 8 × x.
其原理是乘法对加法的分配律：3x 表示 3 × x，因此 3 × x + 5 × x = 8 × x。
Never add the variables together. For instance, x + x equals 2x, not x².
千万不要把字母相加。例如 x + x 等于 2x，而不是 x²。
4. Positive and Negative Terms | 正项与负项
hat term. For 5x - 2x, the second term is -2x, so the calculation is 5 - 2 = 3, giving 3x.
项前面的符号属于该项。对于 5x - 2x，第二项是 -2x，所以计算 5 - 2 = 3，得到 3x。
signs are negative, such as -4y - 3y, combine the coefficients as -4 - 3 = -7, giving -7y.
当两项都是负号时，例如 -4y - 3y，系数合并为 -4 - 3 = -7，得到 -7y。
common mistake is to ignore the negative sign and write -4y - 3y = 7y. That is incorrect.
常见错误是忽略负号，把 -4y - 3y 写成 7y，这是不对的。
5. Terms with Powers and Multiple Variables | 含幂与多个变量的项
With powers, the exponent must match exactly. So 4x² and 6x² can be combined into 10x², but 4x² and 6x³ cannot.
对于含幂的项，指数必须完全相同。因此 4x² 和 6x² 可以合并为 10x²，但 4x² 和 6x³ 不能合并。
For terms with more than one variable, every variable and every exponent must match. For example, 3xy and 7xy are like terms, but 3x²y and 3xy² are not.
对于含有多个变量的项，每个变量和每个指数都必须相同。例如 3xy 和 7xy 是同类项，但 3x²y 和 3xy² 不是。
Multiplication order does not change the term: 2ab and 3ba are like terms because ab = ba.
乘法顺序不会改变项：2ab 和 3ba 是同类项，因为 ab = ba。
6. Step-by-Step Worked Example | 分步示例
Simplify 4x + 3y - 2x + 5y + 7.
化简 4x + 3y - 2x + 5y + 7。
Step 1: Identify like terms. The x terms are 4x and -2x. The y terms are 3y and 5y. The constant is 7.
第 1 步：识别同类项。含 x 的项是 4x 和 -2x；含 y 的项是 3y 和 5y；常数项是 7。
Step 2: Collect the x terms: 4x - 2x = 2x.
第 2 步：合并 x 项：4x - 2x = 2x。
Step 3: Collect the y terms: 3y + 5y = 8y.
第 3 步：合并 y 项：3y + 5y = 8y。
Step 4: Write the simplified expression: 2x + 8y + 7.
第 4 步：写出化简结果：2x + 8y + 7。
The terms 2x, 8y and 7 cannot be combined further because their variable parts are different.
2x、8y 和 7 不能继续合并，因为它们的字母部分不同。
7. Common Mistakes to Avoid | 常见错误
istake 1: Combining unlike terms, such as writing 2x + 3x² = 5x². You cannot add x and x².
错误 1：合并非同类项，例如把 2x + 3x² 写成 5x²。x 和 x² 不能相加。
Mistake 2: Losing the variable, such as writing 3x + 4x = 7 instead of 7x.
错误 2：丢失字母，例如把 3x + 4x 写成 7，而不是 7x。
Mistake 3: Sign errors, such as 6x - 2x = 8x. The correct result is 4x.
错误 3：符号错误，例如 6x - 2x = 8x。正确结果是 4x。
Mistake 4: Changing exponents, such as x + x = x². The correct result is 2x.
错误 4：改变指数，例如 x + x = x²。正确结果是 2x。
8. Collecting Terms After Expanding Brackets | 展开括号后合并同类项
In many IGCSE questions, you must expand brackets first, then collect like terms.
在 IGCSE 的很多题目中，你必须先展开括号，再合并同类项。
Example: Simplify 2(x + 3) + 3(x - 1).
示例：化简 2(x + 3) + 3(x - 1)。
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