📚 Collecting Like Terms: Simplify Algebraic Expressions | 合并同类项：化简代数式
In IGCSE Mathematics, collecting like terms is a core algebra skill. It allows you to simplify expressions before solving equations, substituting values, or rearranging formulas.
在 IGCSE 数学中，合并同类项是核心代数技能。它能帮助你在解方程、代入数值或变换公式之前先化简表达式。
This article explains what like terms are, how to collect them correctly, and how to avoid the most common sign and power mistakes.
本文将解释什么是同类项、如何正确合并同类项，以及如何避免最常见的符号与指数错误。
1. What Are 'Like Terms'? | 什么是同类项
In algebra, an expression is built from terms. A term can be a number, a variable, or numbers and variables multiplied together, such as 7, x, -3y, or 5x².
在代数中，表达式由项组成。项可以是一个数、一个变量，或数与变量相乘，例如 7、x、-3y 或 5x²。
Like terms are terms that have exactly the same variable part and the same power on each variable. For example, 4x and -9x are like terms because both contain x to the power 1.
同类项是指变量部分完全相同且每个变量的指数也相同的项。例如 4x 和 -9x 是同类项，因为两者都含有 x 的一次方。
- Like terms: 3a and -8a | 同类项：3a 和 -8a
- Like terms: 2x² and 5x² | 同类项：2x² 和 5x²
- Like terms: 4xy and -xy | 同类项：4xy 和 -xy
- Not like terms: 3x and 3x² | 非同类项：3x 和 3x²
- Not like terms: 4y and 4y² | 非同类项：4y 和 4y²
- Not like terms: 5ab and 5a | 非同类项：5ab 和 5a
2. Why Collect Like Terms? | 为什么要合并同类项
Collecting like terms makes an expression shorter and easier to use. Instead of writing x + x + x + x, we write 4x.
合并同类项可以让表达式更短、更易使用。与其写 x + x + x + x，不如写 4x。
In IGCSE questions, you are often asked to 'simplify fully'. This usually means expand any brackets first, then collect all like terms so no further combining is possible.
在 IGCSE 题目中，常常要求“完全化简”。这通常意味着先展开括号，再合并所有同类项，直到不能再合并为止。
tituting values also reduces mistakes. If x = 3, it is easier to evaluate 8x than 3x + 5x.
在代入数值前先化简也可以减少错误。如果 x = 3，计算 8x 比计算 3x + 5x 更简单。
3. The Golden Rule: Same Variable, Same Power | 黄金法则：同字母同指数
The golden rule is simple: you can only add or subtract terms when both the variable and its power are identical.
黄金法则很简单：只有当变量及其指数完全相同时，才能相加或相减。
3x + 5x = (3 + 5)x = 8x
Here the coefficients 3 and 5 are added, while the variable part x stays unchanged.
这里系数 3 和 5 相加，而变量部分 x 保持不变。
6y² - 2y² = 4y²
But 3x + 5y cannot be simplified because the variables are different. Also, 3x + 5x² cannot be simplified because the powers are different.
但 3x + 5y 不能化简，因为变量不同。同样，3x + 5x² 也不能化简，因为指数不同。
4. Collecting Positive and Negative Terms | 合并正项与负项
term. For example, in 9a - 12a, the second term is negative, so you calculate 9 - 12 = -3.
合并带符号的项时，要跟随每项前面的符号。例如在 9a - 12a 中，第二项是负项，因此计算 9 - 12 = -3。
9a - 12a = -3a
Use a number line if it helps: adding a negative term means moving left.
如果有帮助，可以使用数轴：加上一个负项意味着向左移动。
For mixed signs, group positive terms and negative terms carefully:
对于混合符号，仔细分组正项和负项：
5m - 3m - 8m + 2m = (5 + 2 - 3 - 8)m = -4m
5. Simplifying Expressions with Multiple Variables | 含多个变量的表达式化简
An expression may contain several different variables. Collect each type separately.
一个表达式可能含有多个不同的变量。需要分别合并每一种类型。
Simplify 7a + 3b - 2a + 5b.
化简 7a + 3b - 2a + 5b。
lect a terms: 7a - 2a = 5a. Collect b terms: 3b + 5b = 8b. The simplified form is 5a + 8b.
合并 a 类项：7a - 2a = 5a。合并 b 类项：3b + 5b = 8b。化简结果为 5a + 8b。
When terms contain two variables, such as xy, treat the whole product as one variable part. So 3xy and -5xy are like terms, but 3xy and 3x are not.
当项含有两个变量时，例如 xy，应将整个乘积视为一个变量部分。因此 3xy 和 -5xy 是同类项，但 3xy 和 3x 不是。
6. Dealing with Powers and Higher Indices | 处理幂与高次项
Remember that x and x² are not like terms. The exponent must match exactly.
记住 x 和 x² 不是同类项。指数必须完全一致。
Simplify 4x² + 3x - x² + 2x - 5.
化简 4x² + 3x - x² + 2x - 5。
t x² terms: 4x² - x² = 3x². Collect x terms: 3x + 2x = 5x. The constant -5 stays as it is.
合并 x² 类项：4x² - x² = 3x²。合并 x 类项：3x + 2x = 5x。常数 -5 保持不变。
4x² + 3x - x² + 2x - 5 = 3x² + 5x - 5
Never combine 3x² and 5x because the powers are different.
切勿合并 3x² 和 5x，因为指数不同。
7. Brackets: Expand First, Then Collect | 括号：先展开，再合并
sent, expand first, then collect like terms. Use the distributive law: a(b + c) = ab + ac.
当出现括号时，先展开，再合并同类项。使用分配律：a(b + c) = ab + ac。
Simplify 3(x + 2) + 4(x - 1).
化简 3(x + 2) + 4(x - 1)。
4. Then collect x terms: 3x + 4x = 7x. Combine constants: 6 - 4 = 2. Final answer: 7x + 2.
展开：3x + 6 + 4x - 4。然后合并 x 项：3x + 4x = 7x。合并常数：6 - 4 = 2。最终答案：7x + 2。
3(x + 2) + 4(x - 1) = 7x + 2
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