📚 Collecting Like Terms | 合并同类项
In IGCSE Mathematics, collecting like terms is one of the first essential algebra skills. It means simplifying an expression by adding or subtracting terms that have exactly the same variable part. This skill is used constantly in solving equations, rearranging formulas, drawing graphs and checking algebraic work.
在 IGCSE 数学中，合并同类项是最基础的代数技能之一。它指的是将变量部分完全相同的项相加或相减，从而化简表达式。这一技能广泛用于解方程、变形公式、绘制图像以及检查代数运算。
1. What Are Algebraic Terms? | 什么是代数项
An algebraic term is a single mathematical object in an expression, separated by plus or minus signs. For example, in the expression 5x + 3y - 2, the terms are 5x, 3y and -2. Each term carries the sign immediately to its left.
代数项是表达式中由加号或减号分隔的单个数学对象。例如，在表达式 5x + 3y - 2 中，项分别是 5x、3y 和 -2。每一项都带有其左侧紧邻的符号。
A term usually contains a coefficient and a variable part. In -4x², the coefficient is -4 and the variable part is x². Constant terms such as 7 or -3 have no variable part at all.
一个项通常包含系数和变量部分。在 -4x² 中，系数是 -4，变量部分是 x²。像 7 或 -3 这样的常数项则完全没有变量部分。
- 5x: coefficient 5, variable part x | 系数 5，变量部分 x
- -3y²: coefficient -3, variable part y² | 系数 -3，变量部分 y²
- 12: constant term | 常数项
2. Recognising Like Terms | 识别同类项
Like terms have exactly the same variable part, including the same variables raised to the same powers. The coefficients can be different. For example, 3x and -5x are like terms, but 3x and 3x² are not like terms because the powers of x are different.
同类项具有完全相同的变量部分，包括相同的变量和相同的幂。系数可以不同。例如，3x 和 -5x 是同类项，但 3x 和 3x² 不是同类项，因为 x 的幂不同。
Terms such as 4ab and -2ba are like terms because multiplication is commutative, so ab and ba represent the same variable part. However, 4ab and 4a are not like terms because one contains the extra variable b.
像 4ab 和 -2ba 这样的项是同类项，因为乘法满足交换律，所以 ab 和 ba 表示相同的变量部分。然而，4ab 和 4a 不是同类项，因为前者多了一个变量 b。
| Terms | 项 | Like terms? | 同类项？ | Reason | 原因 |
|---|---|---|
| 3x and -5x | Yes | 是 | Same variable x | 变量 x 相同 |
| 4ab and -2ba | Yes | 是 | Same variables a and b | 变量 a 和 b 相同 |
| 3x and 3x² | No | 否 | Different powers of x | x 的幂不同 |
| 5xy and 5x | No | 否 | 5xy has an extra variable y | 5xy 多了一个变量 y |
3. The Golden Rule of Collecting Like Terms | 合并同类项的金法则
When collecting like terms, add or subtract only the coefficients and keep the variable part unchanged. This is similar to counting objects of the same type: three pens plus two pens gives five pens, but three pens plus two rulers cannot be combined into one group.
合并同类项时，只加减系数，变量部分保持不变。这类似于对同一类物品进行计数：三支笔加两支笔得到五支笔，但三支笔加两把尺子不能合并成一组。
ax + bx = (a + b)x
three x-values plus two x-values, giving five x-values in total. Therefore, 3x + 2x = 5x.
例如，3x + 2x 表示三个 x 加上两个 x，总共五个 x。因此，3x + 2x = 5x。
You cannot combine terms with different variable parts, because they represent different types of quantities. For example, 3x + 4y is already in its simplest form and cannot be written as a single term.
不能合并变量部分不同的项，因为它们表示不同类型的量。例如，3x + 4y 已经是最简形式，不能写成一项。
4. Collecting Positive and Negative Terms | 合并正项与负项
- 3x = (7 - 3)x = 4x. The subtraction only affects the coefficient, not the variable part.
当涉及负号时，把符号看作系数的一部分。例如，7x - 3x = (7 - 3)x = 4x。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图