Collecting Like Terms | 合并同类项

📚 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

This works because 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

The sign in front of a term belongs to that 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。

When both 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。

A 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

Mistake 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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Collecting Like Terms | 合并同类项