Algebra Mastery: Expressions, Brackets & Equations | 代数精通：表达式、括号与方程

📚 Algebra Mastery: Expressions, Brackets & Equations | 代数精通：表达式、括号与方程

This workbook reviews every core skill from Unit 2 of the Year 8 Higher scheme. Each question is followed by its full model answer, with the key step explained in words and then shown symbolically. Work through it with a notebook: cover the solution, attempt the question yourself, and then compare your method with the one below.

本手册系统复盘八年级高阶（Year 8 Higher）第二单元的全部核心技能。每一道题目之后都给出完整的模型答案，先用文字说明关键步骤，再用数学符号写出过程。建议配合笔记本使用：先遮住答案，独立尝试解题，再把自己的方法与下列解法进行逐项对照。

1. The Language of Algebra | 代数语言

Algebra is a precise language. Before we solve anything, we must name each part of an expression correctly and know exactly what the symbols mean.

代数是一门精准的语言。在解题之前，我们必须准确称呼表达式的各个部分，并明确每一个符号的含义。

• Variable 变量: a letter that represents an unknown number, such as x, n or y. | 表示未知数的字母，如 x、n 或 y。
• Term 项: a single number, a variable, or a product of a number and variable(s), such as 5x² or −3x. | 单独的数、变量，或数与变量的乘积，如 5x² 或 −3x。
• Coefficient 系数: the number multiplying the variable, so in 5x² the coefficient is 5. | 与变量相乘的数，因此 5x² 中系数为 5。
• Constant 常数: a term with no variable, such as +2. | 不含变量的项，例如 +2。
• Expression 表达式: a collection of terms without an equals sign, such as 5x² − 3x + 2. | 不含等号的若干项的组合，如 5x² − 3x + 2。
• Equation 方程: a statement that two expressions are equal, such as 5x − 3 = 12. | 表示两个表达式相等的语句，如 5x − 3 = 12。

Consider the expression 5x² − 3x + 2. Its structure is summarised below.

观察表达式 5x² − 3x + 2，其结构总结如下表。

Part 部分	Name 名称	Meaning 含义
5	coefficient of x² | x² 的系数	the number multiplying x² | 与 x² 相乘的数
x²	quadratic term | 二次项	the variable x, squared | 变量 x 的平方
−3	coefficient of x | x 的系数	the number multiplying x | 与 x 相乘的数
+2	constant | 常数项	a term with no variable | 不含变量的项

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2. Collecting Like Terms | 合并同类项

Like terms have exactly the same variable part. Only the coefficients can change; the letter part must stay identical. We cannot add x to x² because the powers are different.

同类项是指变量部分完全相同的项。合并时只能改变系数，字母部分必须保持不变。我们不能把 x 与 x² 相加，因为它们的幂不同。

Question: Simplify 4a + 7b + a − b.

题目：化简 4a + 7b + a − b。

4a + 7b + a − b = 4a + a + 7b − b = 5a + 6b

We group the a-terms: 4a + a = 5a, and the b-terms: 7b − b = 6b. The model answer is 5a + 6b.

我们将 a 项合并：4a + a = 5a；将 b 项合并：7b − b = 6b。模型答案为 5a + 6b。

Question: Simplify 6x² − 2x − 4x² + 3x.

题目：化简 6x² − 2x − 4x² + 3x。

6x² − 2x − 4x² + 3x = 6x² − 4x² − 2x + 3x = 2x² + x

Note that x is the same as 1x, so 3x − 2x = x. The final expression cannot be simplified further because 2x² and x are not like terms.

注意 x 就是 1x，因此 3x − 2x = x。最终表达式无法再化简，因为 2x² 与 x 不是同类项。

Verification tip: substitute a value, for example a = 2 and b = 1. Left side: 4(2) + 7(1) + 2 − 1 = 16; right side: 5(2) + 6(1) = 16. Both sides agree.

验证技巧：代入一个数值，例如 a = 2、b = 1。左边：4(2) + 7(1) + 2 − 1 = 16；右边：5(2) + 6(1) = 16。两边完全一致。

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3. Multiplying and Dividing Terms | 单项式的乘除

When multiplying terms, multiply the coefficients first, then multiply the variable parts using the index laws. The same logic applies to division.

单项式相乘时，先乘系数，再用指数法则处理变量部分；除法运算同理。

Rule 法则	Example 示例
x² × x³ = x²⁺³	x² × x³ = x⁵
p⁵ ÷ p² = p⁵⁻²	p⁵ ÷ p² = p³
(y²)³ = y²ˣ³	(y²)³ = y⁶

Question: Simplify 3a × 4a.

题目：化简 3a × 4a。

3a × 4a = 3 × 4 × a × a = 12a²

Multiply 3 by 4 to get 12, and multiply a by a to get a². The model answer is 12a².

3 乘 4 得 12，a 乘 a 得 a²。模型答案为 12a²。

Question: Simplify 8m⁶ ÷ 4m².

题目：化简 8m⁶ ÷ 4m²。

8m⁶ ÷ 4m² = (8 ÷ 4) × m⁶⁻² = 2m⁴

Divide 8 by 4 to get 2, and subtract indices: 6 − 2 = 4. Negative coefficients follow the usual sign rules:

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