Page 49 Focus: Solving Linear Equations | 第49页重点：解线性方程

📚 Page 49 Focus: Solving Linear Equations | 第49页重点：解线性方程

This article is designed to help you master the key skill on page 49 of your Cambridge KS3 Maths book: solving linear equations. Whether you are just starting with one-step equations or moving on to two-step problems, this guide will walk you through the balance method, essential techniques, and common pitfalls to watch out for. By the end, you will be able to solve equations confidently and check your answers.

本文旨在帮你掌握 Cambridge KS3 数学教材第49页的核心技能：解线性方程。无论你是刚开始接触一步方程，还是正在学习两步方程，本指南都将带你一步步理解平衡法、基本技巧以及需要避免的常见错误。学完之后，你将能自信地解方程并进行验算。

---

1. What is a Linear Equation? | 什么是线性方程？

A linear equation is an algebraic statement where two expressions are equal, and the unknown variable (often x) is not raised to any power higher than 1. For example, 3x + 2 = 11 is linear, but x² + 1 = 5 is not. In KS3, you only work with linear equations. The goal is to find the value of the variable that makes the equation true.

线性方程是一种代数等式，其中两个表达式相等，且未知数（通常用 x 表示）的最高次数为1。例如 3x + 2 = 11 是线性的，而 x² + 1 = 5 则不是。在 KS3 阶段你只会接触到线性方程。我们的目标是求出使等式成立的未知数的值。

---

2. Understanding the Balance Method | 理解平衡法

Think of an equation as a balanced scale. Whatever you do to one side, you must do exactly the same to the other side to keep it balanced. This is called the balance method. For example, if you add 5 to the left, you must also add 5 to the right. This idea is the key to solving every linear equation.

可以把方程想象成一个平衡的天平。无论你对一边做了什么运算，都必须对另一边做完全相同的运算才能保持平衡。这就是平衡法。例如，如果你在左边加5，右边也必须加5。这个思路是解每一个线性方程的关键。

x + 3 = 7

Equation: x + 3 = 7	方程：x + 3 = 7
Subtract 3 from both sides: x + 3 − 3 = 7 − 3	两边减3：x + 3 − 3 = 7 − 3
Simplify: x = 4	化简得：x = 4

---

3. Solving One-Step Equations (Addition & Subtraction) | 解一步方程（加法与减法）

When an equation involves only one operation, such as addition or subtraction, you can solve it by using the inverse operation. If something is added to x, subtract that number from both sides. If something is subtracted, add it back.

当方程只涉及一种运算，例如加法或减法时，你可以通过逆运算来求解。如果 x 加上了某个数，就从两边减去该数。如果减去了某个数，就加回来。

x + 9 = 15

Subtract 9: x + 9 − 9 = 15 − 9	减9：x + 9 − 9 = 15 − 9
x = 6	x = 6

x − 4 = 10

Add 4 to both sides: x − 4 + 4 = 10 + 4	两边加4：x − 4 + 4 = 10 + 4
x = 14	x = 14

---

4. Solving One-Step Equations (Multiplication & Division) | 解一步方程（乘法与除法）

For multiplication equations like 4x = 20, divide both sides by 4. For division equations like x/5 = 3, multiply both sides by 5. Always use the opposite operation.

对于乘法方程如 4x = 20，两边同时除以4。对于除法方程如 x/5 = 3，两边同时乘以5。始终使用相反的运算。

4x = 20

Divide both sides by 4: 4x ÷ 4 = 20 ÷ 4	两边除以4：4x ÷ 4 = 20 ÷ 4
x = 5	x = 5

x / 5 = 3

Multiply both sides by 5: (x/5) × 5 = 3 × 5	两边乘以5：(x/5) × 5 = 3 × 5
x = 15	x = 15

---

5. Introduction to Two-Step Equations | 引入两步方程

Two-step equations involve two operations. For example, 2x + 3 = 11 has multiplication and addition. To solve, you need to undo the operations in reverse order: deal with addition/subtraction first, then multiplication/division. This is like unpacking a parcel — you remove the outer layer before the inner one.

两步方程涉及两种运算。例如 2x + 3 = 11 包含乘法和加法。求解时，需要按相反的顺序撤销运算：先处理加减，再处理乘除。这就像拆包裹——先拆外层再拆内层。

2x + 3 = 11

Step 1: Subtract 3 from both sides 2x + 3 − 3 = 11 − 3 2x = 8	第1步：两边减3 2x + 3 − 3 = 11 − 3 2x = 8
Step 2: Divide both sides by 2 2x ÷ 2 = 8 ÷ 2 x = 4	第2步：两边除以2 2x ÷ 2 = 8 ÷ 2 x = 4

---

6. Solving Two-Step Equations: Example 1 | 解两步方程：示例1

Let’s solve 3x − 7 = 14. The variable x is multiplied by 3 and then 7 is subtracted. To isolate x, we first undo the subtraction by adding 7 to both sides, then undo the multiplication by dividing by 3.

我们来解 3x − 7 = 14。未知数 x 先乘以3，再减去7。要分离出 x，我们首先通过两边加7来撤销减法，然后通过除以3来撤销乘法。

3x − 7 = 14

Add 7: 3x − 7 + 7 = 14 + 7 → 3x = 21	加7：3x − 7 + 7 = 14 + 7 → 3x = 21
Divide by 3: 3x ÷ 3 = 21 ÷ 3 → x = 7	除以3：3x ÷ 3 = 21 ÷ 3 → x = 7
Check: 3(7) − 7 = 21 − 7 = 14 ✓	检验：3(7) − 7 = 21 − 7 = 14 ✓

---

7. Solving Two-Step Equations: Example 2 | 解两步方程：示例2

Now try an equation with a division: x/4 + 2 = 5. The order of operations says addition comes after division. To solve, we reverse: subtract 2 first, then multiply by 4.

现在尝试一个带除法的方程：x/4 + 2 = 5。根据运算顺序，加法在除法之后进行。求解时我们反过来：先减2，再乘以4。

x/4 + 2 = 5

Subtract 2: x/4 + 2 − 2 = 5 − 2 → x/4 = 3	减2：x/4 + 2 − 2 = 5 − 2 → x/4 = 3
Multiply by 4: (x/4) × 4 = 3 × 4 → x = 12	乘以4：(x/4) × 4 = 3 × 4 → x = 12
Check: 12/4 + 2 = 3 + 2 = 5 ✓	检验：12/4 + 2 = 3 + 2 = 5 ✓

---

8. Dealing with Negative Numbers | 处理负数

Equations with negative numbers require careful handling. Remember the rules: subtracting a negative is the same as adding, and when you multiply or divide by a negative number, the sign of the variable may change. Always apply the balance method — do exactly the same to both sides.

涉及负数的方程需要仔细处理。记住运算法则：减去一个负数等于加上它的相反数；当你乘以或除以负数时，未知数的符号可能会改变。始终坚持平衡法——对两边做完全相同的运算。

−2x = 10

Divide both sides by −2: (−2x) ÷ (−2) = 10 ÷ (−2)	两边除以−2：(−2x) ÷ (−2) = 10 ÷ (−2)
x = −5	x = −5

x − 6 = −2

Add 6 to both sides: x − 6 + 6 = −2 + 6	两边加6：x − 6 + 6 = −2 + 6
x = 4	x = 4

---

9. Equations with Brackets (Intro) | 带括号的方程（简介）

Sometimes equations include brackets, like 3(x + 2) = 15. You can either expand the bracket first or divide both sides by the coefficient. Both methods give the same answer, so choose whichever you find easier.

有时方程含有括号，例如 3(x + 2) = 15。你可以先展开括号，也可以先将两边除以系数。两种方法结果相同，选你觉得更轻松的即可。

Method 1 (Expand): 3x + 6 = 15 → 3x = 9 → x = 3

方法1（展开）：3x + 6 = 15 → 3x = 9 → x = 3

Method 2 (Divide first): (x + 2) = 5 → x = 3

方法2（先除）：(x + 2) = 5 → x = 3

---

10. Common Mistakes to Avoid | 常见错误避免

Mistake 1: Forgetting to apply the operation to both sides. Some students think they can just ‘move’ a number to the other side and change its sign without showing the reasoning. Always write the step clearly: do the same to both sides.

错误1：忘记对两边进行相同运算。有些同学以为只需把数字“移”到另一边并变号，却不写出推理过程。请始终清晰地写出步骤：对两边进行相同操作。

Mistake 2: Doing operations in the wrong order

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）