Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In IGCSE algebra, simplifying expressions is one of the first essential skills. Collecting like terms allows you to rewrite an expression in its shortest form before solving equations, drawing graphs, or substituting values.

在 IGCSE 代数中，化简表达式是最基础的技能之一。合并同类项可以让你在解方程、画图像或代入数值前，先把表达式写成最简形式。

1. What Are Terms in Algebra? | 代数中的项是什么？

In algebra, an expression is made by adding or subtracting terms. A term is a number, a variable, or a product of numbers and variables. For example, 3x, -5y, 7xy, and 4 are separate terms.

在代数中，表达式由项相加或相减组成。项是一个数、一个变量，或数与变量的乘积。例如，3x、-5y、7xy 和 4 都是单独的项。

The + and – signs show where one term ends and the next begins. The sign in front of a term belongs to that term, so 5x – 3 has terms 5x and -3.

加号和减号标明一个项的结束和下一个项的开始。项前面的符号属于该项，因此 5x – 3 的项是 5x 和 -3。

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2. Like Terms and Unlike Terms | 同类项与异类项

Like terms have exactly the same variable part, including the same powers. Unlike terms have different variable parts. Constants are like terms with all other constants.

同类项具有完全相同的变量部分，包括相同的幂。异类项的变量部分不同。常数项与其他所有常数项是同类项。

For example, 3x and -8x are like terms; 4y and 7y are like terms; 2x and 2y are unlike; 3x and 3x² are unlike because the power of x is different.

例如，3x 和 -8x 是同类项；4y 和 7y 是同类项；2x 和 2y 不是同类项；3x 和 3x² 不是同类项，因为 x 的幂不同。

3x and -8x	Like terms: both have x¹
4y and 7y	Like terms: both have y¹
2x and 2y	Unlike terms: different variables
3x and 3x²	Unlike terms: different powers

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3. Why Collecting Means Adding or Subtracting Coefficients | 为什么合并意味着系数相加减

To combine like terms, add or subtract their coefficients. The coefficient is the number part in front of the variable. For example, in 6x, the coefficient is 6.

合并同类项时，将它们的系数相加或相减。系数是变量前面的数字部分。例如，在 6x 中，系数是 6。

3x + 5x = (3 + 5)x = 8x

3x + 5x means (3 + 5)x, so the result is 8x.

3x + 5x 表示 (3 + 5)x，所以结果是 8x。

7y – 2y = (7 – 2)y = 5y

7y – 2y means (7 – 2)y, so the result is 5y.

7y – 2y 表示 (7 – 2)y，所以结果是 5y。

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4. The Sign Rule: Keep the Sign With Its Term | 符号规则：符号随项而走

The sign in front of a term must be kept with its coefficient. For -4a + 9a, add -4 and 9 to get 5a. For -4a – 3a, add -4 and -3 to get -7a.

项前面的符号必须与它的系数一起保留。对于 -4a + 9a，将 -4 和 9 相加得到 5a。对于 -4a – 3a，将 -4 和 -3 相加得到 -7a。

Think of negative coefficients as negative numbers. 5x – 3x – 2x = (5 – 3 – 2)x = 0x = 0.

把负系数看作负数。5x – 3x – 2x = (5 – 3 – 2)x = 0x = 0。

If the coefficient becomes zero, the whole term disappears from the simplified expression.

如果系数变为零，整个项就会从化简后的表达式中消失。

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5. Combining Single-Variable Terms | 合并单一变量项

When a single variable is written alone, its coefficient is 1. For example, x means 1x and -x means -1x.

当单个变量单独写出时，它的系数是 1。例如，x 表示 1x，-x 表示 -1x。

Simplify 7x – 2x + x. This is (7 – 2 + 1)x = 6x.

化简 7x – 2x + x。这是 (7 – 2 + 1)x = 6x。

Simplify -3x + 8x – x. This is (-3 + 8 – 1)x = 4x.

化简 -3x + 8x – x。这是 (-3 + 8 – 1)x = 4x。

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6. Combining Terms with Powers | 合并含幂的项

Terms with powers can be combined only if the variable and its power match exactly. x² and x² are like terms, but x² and x are not like terms.

含幂的项只有在变量及其幂完全相同时才能合并。x² 和 x² 是同类项，但 x² 和 x 不是同类项。

2x² + 5x² – x² = (2 + 5 – 1)x² = 6x²

2x² + 5x² – x² = (2 + 5 – 1)x² = 6x².

2x² + 5x² – x² = (2 + 5 – 1)x² = 6x²。

An expression such as 4x² + 3x + 2x² – x can be simplified to 6x²

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