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 first algebraic tools students meet in Key Stage 3. They appear throughout the Cambridge Lower Secondary Mathematics curriculum and provide the foundation for harder algebra topics later. This article explains how to solve them step by step using the balance method, with clear worked examples and common mistakes to avoid.

线性方程是学生在KS3阶段最早接触的代数工具之一。它们贯穿剑桥初中数学课程，也为以后更难的代数内容打下基础。本文将通过“天平法”分步骤讲解如何解一元一次方程，并配有清晰的例题和需要避免的常见错误。

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

A linear equation is an equation in which the unknown, usually written as x, is raised only to the power of 1. It can appear in forms such as x + 5 = 12, 3x − 4 = 11 or 7 − 2x = 1. The word “linear” tells us that the graph of the equation would be a straight line.

线性方程是指未知数（通常写作 x）的指数只有 1 的方程。它可以表现为 x + 5 = 12、3x − 4 = 11 或 7 − 2x = 1 等形式。“线性”这个词告诉我们，方程的图像会是一条直线。

x + 5 = 12

In this example, the left side x + 5 and the right side 12 are equal. The goal is to find the value of x that makes the statement true.

在这个例子中，左边 x + 5 和右边 12 是相等的。我们的目标是求出使等式成立的 x 的值。

2. The Balance Method | 天平法

Think of an equation as a balanced set of scales. The left side and the right side have the same value. To solve the equation, you must do exactly the same operation to both sides. This keeps the scales balanced and allows you to isolate the unknown.

把方程想象成一架平衡的天平，左边和右边具有相同的值。解方程时，你必须对方程两边做完全相同的运算。这样才能保持天平平衡，并且把未知数单独求出来。

For example, if you subtract 3 from the left side, you must also subtract 3 from the right side. If you divide one side by 2, you must divide the other side by 2 as well. This rule is the foundation of all equation solving.

例如，如果你从左边减去 3，你就必须从右边也减去 3。如果你把一边除以 2，你也必须把另一边除以 2。这条规则是所有方程求解的基础。

3. Solving by Addition and Subtraction | 通过加减法求解

If a number is added to the unknown, subtract that number from both sides. If a number is subtracted from the unknown, add that number to both sides. This undoes the addition or subtraction and helps isolate x.

如果未知数加上了一个数，就在方程两边同时减去这个数。如果未知数减去一个数，就在方程两边同时加上这个数。这样可以消去加法或减法，帮助把 x 单独留在一边。

Example 1: Solve x + 7 = 15.

例 1：解方程 x + 7 = 15。

x + 7 − 7 = 15 − 7

x = 8

Example 2: Solve x − 5 = 9.

例 2：解方程 x − 5 = 9。

x − 5 + 5 = 9 + 5

x = 14

4. Solving by Multiplication and Division | 通过乘除法求解

If the unknown is multiplied by a number, divide both sides by that number. If the unknown is divided by a number, multiply both sides by that number. These inverse operations undo the multiplication or division and reveal the value of x.

如果未知数乘以一个数，就在方程两边同时除以这个数。如果未知数除以一个数，就在方程两边同时乘以这个数。这些逆运算可以消去乘法或除法，从而求出 x 的值。

Example 1: Solve 4x = 28.

例 1：解方程 4x = 28。

4x ÷ 4 = 28 ÷ 4

x = 7

Example 2: Solve x ÷ 3 = 6.

例 2：解方程 x ÷ 3 = 6。

x ÷ 3 × 3 = 6 × 3

x = 18

5. Solving Two-Step Equations | 解两步方程

Many linear equations require two steps to solve. Usually you must first undo the addition or subtraction, and then undo the multiplication or division. Always reverse the normal order of operations. This is sometimes called “undoing BODMAS” or reversing PEMDAS.

许多线性方程需要两步才能解出。通常你必须先消去加法或减法，再消去乘法或除法。一定要逆着正常的运算顺序来做。这有时被称为“倒推 BODMAS”或倒推 PEMDAS。

Example: Solve 2x + 5 = 17.

例题：解方程 2x + 5 = 17。

Step 1: Subtract 5 from both sides.

第一步：方程两边同时减去 5。

2x + 5 − 5 = 17 − 5

2x = 12

Step 2: Divide both sides by 2.

第二步：方程两边同时除以 2。

2x ÷ 2 = 12 ÷ 2

x = 6

6. Equations with Brackets | 含括号的方程

If an equation contains brackets, expand them first using the distributive law. For example, 2(x + 3) means 2 multiplied by x plus 2 multiplied by 3, which gives 2x + 6. After expanding, solve the equation using the balance method.

如果方程含有括号，要先用分配律展开。例如，2(x + 3) 表示 2 乘以 x 加上 2 乘以 3，得到 2x + 6。展开后，再用天平法解方程。

Example: Solve 2(x + 3) = 14.

例题：解方程 2(x + 3) = 14。

2 × x + 2 × 3 = 14

2x + 6 = 14

Now subtract 6 from both sides, then divide by 2.

现在两边同时减去 6，再除以 2。

2x = 8

x = 4

7. Equations with Variables on Both Sides | 两边都含未知数的方程

When the unknown appears on both sides of the equation, collect all the variable terms on one side and all the number terms on the other. A good strategy is to eliminate the smaller variable term first. Then solve the resulting equation as usual.

当未知数出现在方程两边时，要把所有含未知数的项移到一边，把所有数字项移到另一边。一个有效的策略是先把较小的含未知数项消去。然后再按通常方法解方程。

Example: Solve 5x + 2 = 2x + 14.

例题：解方程 5x + 2 = 2x + 14。

Subtract 2x from both sides to remove the smaller variable term.

两边同时减去 2x，以消去较小的含未知数项。

5x − 2x + 2 = 2x − 2x + 14

3x + 2 = 14

Then subtract 2 from both sides and divide by 3.

然后两边同时减去 2，再除以 3。

3x = 12

x = 4

8. Equations with Fractions | 含分数的方程

Equations can also involve fractions. One efficient way to solve them is to multiply every term by the lowest common denominator, often called clearing the fractions. This changes the equation into a simpler form without fractions.

方程也可能含有分数。一个高效的解法是给每一项都乘以最小公分母，通常叫作“去分母”。这样方程就会变成不含分数的更简单形式。

Example: Solve x/2 + x/3 = 5.

例题：解方程 x/2 + x/3 = 5。

The lowest common denominator of 2 and 3 is 6, so multiply every term by 6.

2 和 3 的最小公分母是 6，所以给每一项都乘以 6。

6 × x/2 + 6 × x/3 = 6 × 5

3x + 2x = 30

5x = 30

x = 6

9. Checking Your Answer | 检验答案

Always substitute your solution back into the original equation. The left side and the right side must be equal. Checking your answer catches many common mistakes and gives you confidence in your final result.

一定要把你的解代回原方程。左边和右边必须相等。检验答案能发现许多常见错误，并让你对最终结果更有信心。

Example: Check x = 6 for 2x + 5 = 17.

例题：检验 x = 6 是否满足方程 2x + 5 = 17。

Left side = 2 × 6 + 5 = 12 + 5 = 17

Right side = 17

Since both sides are equal, x = 6 is correct.

因为两边相等，所以 x = 6 是正确的。

10. Common Mistakes | 常见错误

Many students lose marks on linear equations because of small avoidable errors. Read each statement carefully and check your work line by line. Below are some of the most frequent mistakes seen in Key Stage 3 algebra.

许多学生因为一些小的、可以避免的错误而在线性方程上失分。要仔细阅读每一步，并且逐行检查。下面是KS3代数中最常见的一些错误。

Forgetting to apply an operation to both sides of the equation.
Making a sign error when moving a term from one side to the other.
Dividing or multiplying in the wrong order in two-step equations.
Expanding brackets incorrectly, such as writing 2(x + 3) = 2x + 3.
Failing to check the final answer in the original equation.

忘记对方程两边同时进行同一种运算。
把一项从一边移到另一边时搞错正负号。
在两步方程中乘除顺序颠倒。
括号展开错误，例如把 2(x + 3) 写成 2x + 3。
没有把最终答案代回原方程进行检验。

11. Practice Questions | 练习题

Try these quick practice questions before checking the answers. Cover the answer column first and attempt each equation using the balance method.

在查看答案之前，先试做以下快速练习题。可以先遮住答案栏，用天平法尝试解每一个方程。

Question \| 题目	Answer \| 答案
x + 9 = 14	x = 5
3x − 7 = 11	x = 6
2(x + 4) = 18	x = 5
4x + 3 = 2x + 15	x = 6
x/3 + 2 = 6	x = 12
5 − 2x = 1	x = 2

12. Summary | 总结

To solve a linear equation confidently, keep the equation balanced at every step, undo operations in reverse order, expand brackets first, collect variable terms on one side, and always check your final answer by substitution. With regular practice, these steps become automatic and prepare you well for more advanced algebra.

要自信地解一元一次方程，就要在每一步都保持方程平衡，按逆运算顺序消去运算，先展开括号，把含未知数的项移到一边，并且始终通过代入来检验最终答案。通过经常练习，这些步骤会变得自动化，为你学习更高级的代数做好准备。

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Solving Linear Equations | 解一元一次方程