Algebraic Expressions and Linear Equations | 代数式与一元一次方程

📚 Algebraic Expressions and Linear Equations | 代数式与一元一次方程

This revision guide covers the essential KS3 Cambridge Mathematics skills for working with algebraic expressions and solving linear equations. You will learn how to simplify expressions, expand brackets, solve equations step by step, and apply these skills to word problems.

本复习指南涵盖 KS3 剑桥数学中代数式与一元一次方程的核心技能。你将学习如何化简代数式、去括号、逐步解方程，并将这些技能应用于文字题。

1. Algebraic Language and Notation | 代数语言与符号

le 2x + 5. An equation is a statement that two expressions are equal, such as 2x + 5 = 11.

在代数中，x、y 或 n 等字母用来表示未知数。项是单个数字、字母或乘积，例如 4x、−3y 或 7。代数式是由 + 或 − 连接项组合而成，例如 2x + 5。方程是说明两个代数式相等的式子，例如 2x + 5 = 11。

The number in front of a letter is called the coefficient. In 4x, the coefficient is 4. A constant is a term with no letter, such as 5 or −2.

字母前面的数字称为系数。在 4x 中，系数是 4。常数是不含字母的项，例如 5 或 −2。

Term	Meaning	中文含义
Coefficient	number in front of a letter	字母前的系数
Constant	number on its own	常数
Variable	letter that stands for an unknown	变量

Understanding these basic words helps you follow algebraic arguments and explain your working clearly in tests.

理解这些基本术语有助于你在考试中理解代数推理，并清楚地说明解题过程。

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2. Collecting Like Terms | 合并同类项

Like terms contain exactly the same letters raised to the same powers. For example, 3x and 5x are like terms, while 3x and 3x² are not. To simplify an expression, add or subtract the coefficients of like terms.

同类项含有完全相同的字母且次数相同。例如，3x 和 5x 是同类项，而 3x 和 3x² 不是。化简代数式时，把同类项的系数相加或相减。

e x terms: 4x − 2x = 2x. Collect the y terms: 3y + y = 4y. The simplified form is 2x + 4y.

化简 4x + 3y − 2x + y。合并 x 项：4x − 2x = 2x。合并 y 项：3y + y = 4y。化简结果是 2x + 4y。

4x + 3y − 2x + y = 2x + 4y

Always keep the sign with each term when rearranging. A common mistake is to lose a negative sign or to combine unlike terms.

重新排列时，每个项都要带着它前面的符号。常见错误是丢失负号或合并不同类项。

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3. Expanding Brackets | 去括号展开

term inside the bracket by the term outside. Use the distributive law: a(b + c) = ab + ac.

去括号展开是指用括号外的项乘以括号内的每一项。使用分配律：a(b + c) = ab + ac。

Expand 3(x + 2). Multiply 3 by x to get 3x, and 3 by 2 to get 6. The result is 3x + 6.

展开 3(x + 2)。3 乘以 x 得到 3x，3 乘以 2 得到 6。结果是 3x + 6。

3(x + 2) = 3x + 6

With a negative outside, be careful: −2(x − 5) = −2 × x + (−2) × (−5) = −2x + 10.

括号外是负数时要小心：−2(x − 5) = −2 × x + (−2) × (−5) = −2x + 10。

−2(x − 5) = −2x + 10

Expanding correctly is essential before solving equations that contain brackets.

在解含有括号的方程之前，正确展开是必不可少的。

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4. Solving One-Step Equations | 解一步方程

To solve an equation, find the value of the unknown that makes the equation true. For one-step equations, apply the inverse operation to both sides.

解方程就是求出使等式成立的未知数的值。对于一步方程，在等式两边同时进行逆运算。

Solve x + 3 = 7. Subtract 3 from both sides: x + 3 − 3 = 7 − 3, so x = 4.

解 x + 3 = 7。两边同时减去 3：x + 3 − 3 = 7 − 3，所以 x = 4。

x + 3 = 7 ⇒ x = 4

Solve 5x = 20. Divide both sides by 5: 5x ÷ 5 = 20 ÷ 5, so x = 4.

解 5x = 20。两边同时除以 5：5x ÷ 5 = 20 ÷ 5，所以 x = 4。

5x = 20 ⇒ x = 4

The equation stays balanced only when you do exactly the same operation to both sides.

只有对等式两边同时进行完全相同的运算，方程才能保持平衡。

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5. Solving Two-Step Equations | 解两步方程

A two-step equation has two operations applied to the unknown. Undo addition or subtraction first, then undo multiplication or division.

两步方程中，未知数经历了两步运算。先消去加减法，再消去乘除法。

2x + 3 = 11. First subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4.

解 2x + 3 = 11。首先两边同时减去 3：2x = 8。然后两边同时除以 2：x = 4。

2x + 3 = 11 ⇒ 2x = 8 ⇒ x = 4

For 5x − 4 = 16, add 4 to both sides to get 5x = 20, then divide by 5 to get x = 4.

对于 5x − 4 = 16，两边同时加上 4 得到 5x = 20，再除以 5 得到 x = 4。

5x − 4 = 16 ⇒ 5x = 20 ⇒ x = 4

Following the correct order of inverse operations prevents many mistakes in longer equations.

按照正确的逆运算顺序解题，可以避免在较长方程中出错。

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6. Equations with Brackets | 含括号的方程

If an equation contains brackets, expand them first if necessary. Then simplify both sides before solving.

如果方程中含有括号，必要时先展开。然后在解方程前化简两边。

3(x + 2) = 15. Expand the left side: 3x + 6 = 15. Subtract 6: 3x = 9. Divide by 3: x = 3.

解 3(x + 2) = 15。展开左边：3x + 6 = 15。减去 6：3x = 9。除以 3：x = 3。

3(x + 2) = 15 ⇒ 3x + 6 = 15 ⇒ x = 3

divide first: 3(x + 2) = 15 means x + 2 = 5 when both sides are divided by 3. Then x = 3.

有时可以先除：3(x + 2) = 15 表示两边除以 3 后 x + 2 = 5。然后 x = 3。

3(x + 2) = 15 ⇒ x + 2 = 5 ⇒ x = 3

Choosing an efficient first step makes your working shorter and clearer.

选择高效的第一步可以让解题过程更简短、更清晰。

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7. Equations with Unknowns on Both Sides | 未知数在等式两边的方程

When the unknown appears on both sides of the equation, collect the variable terms on one side and the constants on the other.

当未知数出现在等式两边时，把含未知数的项移到一边，常数移到另一边。

+ 8. Subtract 3x from both sides: 2x + 2 = 8. Then subtract 2: 2x = 6. Divide by 2: x = 3.

解 5x + 2 = 3x + 8。两边同时减去 3x：2x + 2 = 8。再减去 2：2x = 6。除以 2：x = 3。

5x + 2 = 3x + 8 ⇒ 2x = 6 ⇒ x = 3

Always keep the equation balanced by doing the same operation to both sides. Check that the sign moves correctly when subtracting terms.

始终在等式两边同时进行相同运算以保持平衡。移项时注意符号是否正确。

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8. Using Inverse Operations | 使用逆运算

Every equation-solving step relies on inverse operations: addition and subtraction are inverses, and multiplication and division are inverses. Think of a function machine running backwards.

解方程的每一步都依赖逆运算：加法与减法互为逆运算，乘法与除法互为逆运算。可以想象一台函数机器反向运行。

forward machine is: x → ×2 → +3 → 11. The reverse machine is: 11 → −3 → ÷2 → 4, so x = 4.

对于 2x + 3 = 11，正向机器是：x → ×2 → +3 → 11。反向机器是：11 → −3 → ÷2 → 4，所以 x = 4。

Forward: x → 2x → 2x + 3 = 11

Reverse: 11 − 3 = 8, 8 ÷ 2 = 4

Equation	First inverse operation	Second inverse operation
2x + 3 = 11	subtract 3	divide by 2
5x − 4 = 16	add 4	divide by 5
x/3 + 2 = 6	subtract 2	multiply by 3

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9. Forming Equations from Word Problems | 由文字题列方程

Read the problem carefully and choose a letter for the unknown. Translate the words into an equation using the operations described.

仔细阅读题目，并用一个字母表示未知数。根据描述的运算将文字转化为方程。

Example: I think of a number, multiply it by 3 and add 4. The result is 19. Let the number be n, so 3n

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