Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

In this article, we will learn how to solve linear equations in one variable. This is one of the most important skills in algebra, and it is also a key topic in the IGCSE Mathematics syllabus.

在本文中，我们将学习如何解一元一次方程。这是代数中最重要的技能之一，同时也是 IGCSE 数学考纲中的关键内容。

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

A linear equation in one variable is an equation that can be written in the form ax + b = c, where x is the variable, and a, b, c are constants, with a ≠ 0. The term “linear” means the variable is raised to the power of 1 only.

一元一次方程是指可以写成 ax + b = c 形式的方程，其中 x 是变量，a、b、c 是常数，且 a ≠ 0。这里“一次”的意思是变量的最高次数仅为 1。

For example:

例如：

3x + 2 = 11

This equation is linear because the variable x appears only to the first power.

这个方程是线性的，因为变量 x 只以一次方出现。

2. Keeping Equations Balanced | 保持方程平衡

An equation is like a balance scale. The left-hand side and the right-hand side are equal. If you change one side in a certain way, you must change the other side in the exact same way to keep the balance.

方程就像一架天平。左边和右边相等。如果你对一边做了某种改变，就必须对另一边做完全相同的改变，才能保持平衡。

If you add a number to one side, add the same number to the other side.
如果你给一边加上一个数，也要给另一边加上同一个数。
If you subtract a number from one side, subtract the same number from the other side.
如果你从一边减去一个数，也要从另一边减去同一个数。
If you multiply or divide one side, multiply or divide the other side by the same non-zero number.
如果你对一边进行乘法或除法，就要对另一边乘以或除以同一个非零数。

This principle is the foundation of solving equations.

这个原则是解方程的基础。

3. Inverse Operations | 逆运算

To solve an equation, we use inverse (opposite) operations to isolate the variable. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

解方程时，我们用逆运算（相反运算）来把变量单独分离出来。加法和减法互为逆运算；乘法和除法互为逆运算。

Operation	Inverse Operation
Addition (+) 加法	Subtraction (−) 减法
Subtraction (−) 减法	Addition (+) 加法
Multiplication (×) 乘法	Division (÷) 除法
Division (÷) 除法	Multiplication (×) 乘法

For example, to undo adding 5, we subtract 5. To undo multiplying by 3, we divide by 3.

例如，要抵消加 5，我们就减 5；要抵消乘以 3，我们就除以 3。

4. Solving Equations of the Form ax + b = c | 解形如 ax + b = c 的方程

Let us solve the equation 2x + 3 = 7 step by step.

我们一步一步来解方程 2x + 3 = 7。

Step 1: Subtract 3 from both sides to keep the balance.

步骤 1：两边同时减去 3，以保持平衡。

2x + 3 − 3 = 7 − 3

2x = 4

Step 2: Divide both sides by 2 to isolate x.

步骤 2：两边同时除以 2，以分离出 x。

2x ÷ 2 = 4 ÷ 2

x = 2

So the solution is x = 2.

所以解为 x = 2。

5. Equations with Variables on Both Sides | 解含有未知数两边的方程

Some equations have the variable on both sides. For example: 5x − 2 = 3x + 8.

有些方程两边都含有未知数，例如：5x − 2 = 3x + 8。

Step 1: Subtract 3x from both sides to gather the variable terms on the left.

步骤 1：两边同时减去 3x，把含未知数的项集中到左边。

5x − 3x − 2 = 3x − 3x + 8

2x − 2 = 8

Step 2: Add 2 to both sides.

步骤 2：两边同时加上 2。

2x = 10

Step 3: Divide both sides by 2.

步骤 3：两边同时除以 2。

x = 5

The solution is x = 5.

解为 x = 5。

6. Equations with Brackets | 含括号的方程

When an equation contains brackets, we first expand the brackets using the distributive property, then solve as usual.

当方程中含有括号时，我们先用分配律展开括号，然后按常规步骤求解。

For example: 3(x + 4) = 21.

例如：3(x + 4) = 21。

Step 1: Expand the bracket.

步骤 1：展开括号。

3 × x + 3 × 4 = 21

3x + 12 = 21

Step 2: Subtract 12 from both sides.

步骤 2：两边同时减去 12。

3x = 9

Step 3: Divide both sides by 3.

步骤 3：两边同时除以 3。

x = 3

Always expand brackets before applying inverse operations on the variable.

在对方程中的变量使用逆运算之前，一定要先展开括号。

7. Word Problems with Equations | 用方程解应用题

Linear equations are often used to solve word problems. We need to translate the words into an equation by choosing a letter for the unknown value.

一元一次方程常被用来解决应用题。我们需要选择一个字母代表未知数，并把文字翻译成一个方程。

Example: “A number multiplied by 4 and increased by 5 equals 29. Find the number.”

例题：“一个数乘以 4 后再加 5 等于 29。求这个数。”

Let the number be x. Then the equation is:

设这个数为 x。那么方程为：

4x + 5 = 29

Subtract 5 from both sides:

两边同时减去 5：

4x = 24

Divide by 4:

两边同时除以 4：

x = 6

The number is 6.

这个数是 6。

8. Checking Solutions | 验证解

After solving an equation, always check the solution by substituting it back into the original equation. This helps to catch errors.

解完方程后，一定要把解代回原方程进行验证。这有助于发现错误。

Let us check x = 5 for 5x − 2 = 3x + 8.

我们验证 5x − 2 = 3x + 8 的解 x = 5。

Left-hand side: 5 × 5 − 2 = 25 − 2 = 23.

左边：5 × 5 − 2 = 25 − 2 = 23。

Right-hand side: 3 × 5 + 8 = 15 + 8 = 23.

右边：3 × 5 + 8 = 15 + 8 = 23。

Both sides are equal, so the solution is correct.

两边相等，所以解是正确的。

9. Common Mistakes | 常见错误

Students often make the following mistakes when solving linear equations. Be careful to avoid them.

学生在解一元一次方程时经常犯以下错误，请务必小心避免。

Forgetting to do the same operation on both sides. Always keep the equation balanced.
忘记两边做相同的运算。始终要保持方程平衡。
Sign errors when moving terms. If you subtract a term, the sign should change correctly.
移项时符号错误。减去一项时，符号要正确变化。
Not expanding brackets correctly. Multiply every term inside the bracket.
括号展开错误。必须乘以括号内的每一项。
Dividing only one term instead of the entire side. Divide the whole expression on each side.
只除以一边的某一项而不是整个式子。要对每一边的整体进行除法。

10. Practice Questions | 练习题

Try solving the following equations on your own. The answers are provided below.

请自己尝试解以下方程。参考答案已经附在下方。

Question 题目	Answer 答案
1. 2x + 5 = 13	x = 4
2. 6x − 7 = 5x + 3	x = 10
3. 4(x − 2) = 16	x = 6
4. 3x + 2 = 2x + 8	x = 6
5. 5 − 2x = 1	x = 2

If you got all of them correct, congratulations! If not, review the steps and try again.

如果你全部做对了，恭喜你！如果没有全对，请复习一下步骤，再试一次。

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Solving Linear Equations | 解一元一次方程