Unit 3 Algebra: Expressions, Equations and Formulae – Answer Guide | 数学第3单元：代数式、方程与公式答案指南

📚 Unit 3 Algebra: Expressions, Equations and Formulae – Answer Guide | 数学第3单元：代数式、方程与公式答案指南

This answer guide covers the essential skills in Unit 3: algebraic expressions, linear equations, inequalities, formulae, and sequences. Each section gives a clear method, a worked example, and a common mistake to avoid.

本答案指南涵盖第3单元的核心技能：代数式、线性方程、不等式、公式和数列。每一小节都给出清晰方法、范例演示和需要避免的常见错误。

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1. Algebraic Expressions and Terms | 代数式与项

An algebraic expression is a combination of numbers, letters, and operations. Terms are separated by + or − signs. For example, in 5x + 3y − 7, the terms are 5x, 3y, and 7.

代数式是数字、字母和运算符号的组合。项由加号或减号分隔。例如，在 5x + 3y − 7 中，项分别是 5x、3y 和 7。

A coefficient is the number multiplying a variable. In 5x, the coefficient is 5; in −3y, the coefficient is −3. A constant is a term with no variable, such as 7.

系数是乘以变量的数。在 5x 中，系数是 5；在 −3y 中，系数是 −3。常数是不含变量的项，例如 7。

Common mistake: forgetting the sign in front of a term. Always take the sign before the term as part of that term.

常见错误：忽略项前面的符号。始终把项前面的符号看作该项的一部分。

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2. Collecting Like Terms | 合并同类项

Like terms have exactly the same variable part. You can add or subtract like terms by combining their coefficients. Unlike terms cannot be combined.

同类项具有完全相同的变量部分。你可以通过合并系数来加减同类项。不同类项不能合并。

Worked example: simplify 3x + 5y − 2x + 7y.

范例：化简 3x + 5y − 2x + 7y。

3x + 5y − 2x + 7y = (3 − 2)x + (5 + 7)y = x + 12y

The x terms are combined and the y terms are combined. The answer cannot be simplified further because x and y are unlike terms.

把 x 项合并，把 y 项合并。答案不能进一步化简，因为 x 和 y 是不同类项。

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3. Expanding Brackets | 去括号展开

tside by every term inside the bracket. This uses the distributive law a(b + c) = ab + ac.

去括号时，用括号外的项乘以括号内的每一项。这使用了分配律 a(b + c) = ab + ac。

Worked example: expand 3(2x − 4).

范例：展开 3(2x − 4)。

3(2x − 4) = 3 × 2x + 3 × (−4) = 6x − 12

rms and then collect like terms. Example: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.

对于双括号，逐对展开各项，再合并同类项。示例：(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6。

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4. Factorising Expressions | 因式分解

Factorising is the reverse of expanding. Find the highest common factor (HCF) of all terms and place it outside the bracket.

因式分解是去括号的逆运算。找出所有项的最大公因数（HCF），并将其放在括号外。

Worked example: factorise 6x + 9.

范例：因式分解 6x + 9。

6x + 9 = 3(2x + 3)

Check by expanding: 3(2x + 3) = 6x + 9. Always check your factorisation by expanding back.

通过展开检验：3(2x + 3) = 6x + 9。因式分解后一定要展开回去检验。

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5. Solving Linear Equations | 解一元一次方程

A linear equation has the variable to the power of 1. To solve it, use inverse operations to isolate the variable on one side.

一元一次方程的变量的次数为 1。解方程时，使用逆运算将变量孤立在等式一边。

Worked example: solve 2x + 3 = 11.

范例：解方程 2x + 3 = 11。

2x + 3 = 11
2x = 8
x = 4

th sides by 2. The solution is x = 4. Check by substitution: 2(4) + 3 = 11, which is true.

步骤1：两边同时减去 3。步骤2：两边同时除以 2。解为 x = 4。代入检验：2(4) + 3 = 11，成立。

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6. Equations with Unknowns on Both Sides | 两边均为未知数的方程

If the variable appears on both sides, collect variable terms on one side and constants on the other side.

如果变量同时出现在等式两边，把变量项移到一边，常数项移到另一边。

Worked example: solve 5x + 2 = 3x + 8.

范例：解方程 5x + 2 = 3x + 8。

5x + 2 = 3x + 8
5x − 3x = 8 − 2
2x = 6
x = 3

Subtract 3x from both sides, then subtract 2 from both sides,

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