📚 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。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图