Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are one of the most important algebra topics in the Cambridge KS3 mathematics curriculum. They appear in almost every Checkpoint paper, either as standalone questions or hidden inside word problems, sequences and geometry questions. Building a confident, step-by-step method now will save time in Papers 1 and 2 and prepare you for IGCSE algebra.

一元一次方程是剑桥 KS3 数学课程中最重要的代数主题之一。它们几乎出现在每一份 Checkpoint 试卷中，要么作为独立题目，要么隐藏在应用题、数列和几何问题里。现在建立自信、按步骤解题的方法，能让你在 Paper 1 和 Paper 2 中节省时间，并为 IGCSE 代数做好准备。

This revision guide covers the balance method, one-step and two-step equations, brackets, fractions, unknowns on both sides, checking, word problems and exam-style tips. Each section gives you an English explanation followed by a matching Chinese explanation so you can revise bilingually.

本复习指南涵盖天平法、一步和两步方程、括号、分数、等号两边有未知数、检验、应用题和考试技巧。每一节先给出英文讲解，再给出对应的中文讲解，方便你进行双语复习。

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1. What Is a Linear Equation? | 什么是一元一次方程？

A linear equation is an algebraic statement that two expressions are equal. In KS3, the unknown is usually written as x, y, n or a, and the highest power of the unknown is 1. This is why the equation is called ‘linear’: its graph is a straight line.

一元一次方程是一个代数等式，表示两个表达式相等。在 KS3 中，未知数通常写作 x、y、n 或 a，并且未知数的最高次数为 1。这就是为什么它被称为“线性”方程：它的图像是一条直线。

The general form is ax + b = c, where a, b and c are known numbers and a is not zero. For example, 2x + 3 = 11 has a = 2, b = 3 and c = 11.

一般形式是 ax + b = c，其中 a、b 和 c 是已知数，并且 a 不等于 0。例如，2x + 3 = 11 中 a = 2，b = 3，c = 11。

Example: x + 3 = 7 is linear, but x² + 3 = 7 is not linear because the power of x is 2.

示例：x + 3 = 7 是一元一次方程，但 x² + 3 = 7 不是一元一次方程，因为 x 的次数是 2。

The goal is always to find the value of the unknown that makes the equation true. That value is called the solution or root of the equation.

目标始终是找到使方程成立的未知数的值。这个值叫做方程的解或根。

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2. The Balance Method | 天平法（等式的平衡性质）

Think of an equation as a balanced scale. The expression on the left must equal the expression on the right. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side to keep the scale balanced.

把方程想象成一台平衡的天平。左边的表达式必须等于右边的表达式。如果你对一边加、减、乘或除以一个数，你必须对另一边做完全相同的运算，才能保持天平平衡。

Adding the same number to both sides keeps the equation true. For example, if x = 5, then x + 3 = 5 + 3.

两边同时加上同一个数，等式仍然成立。例如，如果 x = 5，那么 x + 3 = 5 + 3。

Subtracting the same number from both sides keeps the equation true. For example, if x = 5, then x – 2 = 5 – 2.

两边同时减去同一个数，等式仍然成立。例如，如果 x = 5，那么 x – 2 = 5 – 2。

Multiplying or dividing both sides by the same non-zero number also keeps the equation true. For example, if x = 5, then 3x = 15 and x ÷ 5 = 1.

两边同时乘以或除以同一个非零数，等式仍然成立。例如，如果 x = 5，那么 3x = 15，并且 x ÷ 5 = 1。

This balance method is the foundation of all equation solving. Once you understand it, harder equations simply become a series of balanced moves.

天平法是解所有方程的基础。一旦你理解了它，更复杂的方程就只是一系列保持平衡的运算步骤。

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3. Solving One-Step Equations | 解一步方程

One-step equations need only one operation to isolate the unknown. The key is to apply the inverse operation on both sides of the equation.

一步方程只需一步运算就能将未知数单独留在等号一边。关键是对方程两边同时进行逆运算。

x + 5 = 12

x = 12 – 5

x = 7

For subtraction, addition is the inverse operation: x – 4 = 9 becomes x = 9 + 4 = 13.

对于减法，加法是逆运算：x – 4 = 9 变为 x = 9 + 4 = 13。

For multiplication, division is the inverse operation: 3x = 18 becomes x = 18 ÷ 3 = 6.

对于乘法，除法是逆运算：3x = 18 变为 x = 18 ÷ 3 = 6。

For division, multiplication is the inverse operation: x ÷ 2 = 7 becomes x = 7 × 2 = 14.

对于除法，乘法是逆运算：x ÷ 2 = 7 变为 x = 7 × 2 = 14。

When you write your solution, keep the equals signs lined up vertically. This makes your

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