Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are at the heart of Key Stage 3 algebra. They are used to describe number patterns, solve real-life problems and prepare for more advanced work with graphs and simultaneous equations. In this revision guide you will learn how to solve equations step by step using the balancing method, how to check your solution and how to avoid common errors.

线性方程是 KS3 代数阶段的核心内容。它们用于描述数字规律、解决现实生活中的问题，并为后续的函数图像与联立方程学习打下基础。在本复习指南中，你将学习如何用等式平衡法逐步解方程、如何检验解以及如何避免常见错误。

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1. What Is an Equation? | 什么是方程？

An equation states that two expressions are equal. The left side must have the same value as the right side. The unknown is usually written as a letter such as x or y. For example, x + 3 = 8 is an equation because it says that when 3 is added to x, the result is 8.

方程表示两个表达式相等。左边必须与右边具有相同的值。未知数通常用字母如 x 或 y 表示。例如，x + 3 = 8 是一个方程，因为它表示 x 加 3 的结果是 8。

The solution of an equation is the value of the unknown that makes the equation true. For x + 3 = 8, the solution is x = 5 because 5 + 3 = 8. You can check this by substituting 5 for x in the original equation.

方程的解是使方程成立的未知数的值。对于 x + 3 = 8，解为 x = 5，因为 5 + 3 = 8。你可以将 5 代入原方程来检验这一点。

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2. The Balancing Method | 等式平衡法

Think of an equation as a balanced set of scales. Whatever operation you do to one side, you must also do to the other side to keep the scales balanced. This is called the balancing method.

把方程想象成一组平衡的天平。你对一边进行的任何运算，也必须对另一边进行同样的运算，以保持天平平衡。这就是等式平衡法。

For example, to solve x + 4 = 10, subtract 4 from both sides:

例如，解 x + 4 = 10，两边同时减去 4：

x + 4 – 4 = 10 – 4, so x = 6

The table below lists common operations and their inverse operations.

下表列出了常见运算及其逆运算。

Operation	Inverse operation
Add 5	Subtract 5
Subtract 7	Add 7
Multiply by 3	Divide by 3
Divide by 4	Multiply by 4

Always apply the inverse operation to both sides of the equation.

一定要对等式的两边同时进行逆运算。

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3. One-Step Equations | 一步方程

One-step equations need only one operation to isolate the variable. Addition and subtraction are inverse operations; multiplication and division are inverse operations.

一步方程只需要一次运算就能分离出变量。加法与减法是逆运算；乘法与除法也是逆运算。

Example 1: Solve x – 7 = 12. Add 7 to both sides:

例题 1：解 x – 7 = 12。两边同时加 7：

x – 7 + 7 = 12 + 7, so x = 19

Example 2: Solve 4x = 20. Divide both sides by 4:

例题 2：解 4x = 20。两边同时除以 4：

4x ÷ 4 = 20 ÷ 4, so x = 5

Example 3: Solve x ÷ 3 = 6. Multiply both sides by 3:

例题 3：解 x ÷ 3 = 6。两边同时乘以 3：

x ÷ 3 × 3 = 6 × 3, so x = 18

Notice that the variable is always left on one side after the inverse operation is applied.

注意，应用逆运算后，变量总是被留在方程的一边。

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4. Two-Step Equations | 两步方程

Two-step equations need two inverse operations. Always undo the addition or subtraction first, then undo the multiplication or division.

两步方程需要两次逆运算。一定要先消去加法或减法，再消去乘法或除法。

Example 1: Solve 2x + 3 = 11.

例题 1：解 2x + 3 = 11。

Step 1: subtract 3 from both sides: 2x + 3 – 3 = 11 – 3, so 2x = 8.

步骤 1：两边同时减 3：2x + 3 – 3 = 11 – 3，得 2x = 8。

Step 2: divide both sides by 2: 2x ÷ 2 = 8 ÷ 2, so x = 4.

步骤 2：两边同时除以 2：2x ÷ 2 = 8 ÷ 2，所以 x = 4。

Example 2: Solve x/5 – 2 = 6.

例题 2：解 x/5 – 2 = 6。

Step 1: add 2 to both sides: x/5 – 2 + 2 = 6 + 2, so x/5 = 8.

步骤 1：两边同时加 2：x/5 – 2 + 2 = 6 + 2，得 x/5 = 8。

Step 2: multiply both sides by 5: x/5 × 5 = 8 × 5, so x = 40.

步骤 2：两边同时乘以 5：x/5 × 5 = 8 × 5，所以 x = 40。

Always keep your working tidy so you can find errors quickly.

始终保持书写工整，这样你可以快速发现错误。

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5. Equations with Brackets | 含括号的方程

If an equation contains brackets, expand them first using the distributive law, then solve using the balancing method.

如果方程含有括号，先用分配律展开括号，然后用等式平衡法求解。

Example: Solve 3(x + 2) = 21.

例题：解 3(x + 2) = 21。

Expand: 3 × x + 3 × 2 = 21, so 3x + 6 = 21.

展开：3 × x + 3 × 2 = 21，得 3x + 6 = 21。

Subtract 6 from both sides: 3x + 6 – 6 = 21 – 6, so 3x = 15.

两边同时减 6：3x + 6 – 6 = 21 – 6，得 3x = 15。

Divide both sides by 3: 3x ÷ 3 = 15 ÷ 3, so x = 5.

两边同时除以 3：3x ÷ 3 = 15 ÷ 3，所以 x = 5。

Alternative method: divide both sides by 3 first: x + 2 = 7, then subtract 2: x = 5. Both methods are correct.

另一种方法：先两边同时除以 3：x + 2 = 7，再减 2：x = 5。两种方法都是正确的。

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6. Equations with Unknowns on Both Sides | 两边含未知数的方程

When both sides of an equation contain the unknown, collect the unknown terms on one side and the number terms on the other side.

当方程两边都含有未知数时，把含未知数的项移到一边，把数字项移到另一边。

Example: Solve 5x – 3 = 2x + 9.

例题：解 5x – 3 = 2x + 9。

Subtract 2x from both sides: 5x – 2x – 3 = 2x – 2x + 9, so 3x – 3 = 9.

两边同时减 2x：5x – 2x – 3 = 2x – 2x + 9，得 3x – 3 = 9。

Add 3 to both sides: 3x – 3 + 3 = 9 + 3, so 3x = 12.

两边同时加 3：3x – 3 + 3 = 9 + 3，得 3x = 12。

Divide both sides by 3: 3x ÷ 3 = 12 ÷ 3, so x = 4.

两边同时除以 3：3x ÷ 3 = 12 ÷ 3，所以 x = 4。

Always move the smaller unknown term to the larger one to keep the coefficient positive.

总是把较小的未知数项移到较大的未知数项一边，使系数保持为正。

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7. Equations with Fractions | 含分数的方程

To solve an equation with fractions, multiply every term on both sides by the lowest common denominator to clear the fractions.

解含分数的方程时，将两边每一项乘以最小公分母，以消去分数。

Example: Solve x/3 + 1 = 5/6.

例题：解 x/3 + 1 = 5/6。

The lowest common denominator of 3 and 6 is 6. Multiply every term by 6:

3 和 6 的最小公分母为 6。将每一项乘以 6：

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