📚 Solving Linear Equations | 一元一次方程求解
Linear equations are one of the most important building blocks in KS3 mathematics. They appear in algebra, word problems, graphs and even in science formulas. This article explains how to solve linear equations step by step, from simple one-step equations to equations with brackets and unknowns on both sides.
一元一次方程是 KS3 数学中最重要的基础内容之一。它们出现在代数、应用题、图像甚至科学公式中。本文将逐步讲解如何解一元一次方程，从简单的一步方程到含括号以及两边均含未知数的方程。
1. What Is a Linear Equation? | 什么是一元一次方程？
A linear equation is an equation where the unknown variable has a power of 1. The general form is ax + b = c, where a, b and c are numbers. There are no squared terms such as x² or cubed terms such as x³.
一元一次方程是指未知数的最高次数为 1 的方程。其一般形式为 ax + b = c，其中 a、b、c 是数。方程中不能出现 x² 或 x³ 这样的高次项。
Examples of linear equations include x + 5 = 12, 3y − 4 = 11 and 2x + 1 = x + 6.
线性方程的例子包括 x + 5 = 12、3y − 4 = 11 和 2x + 1 = x + 6。
2. The Balance Method | 天平法
Think of an equation as a balanced scale. If you add, subtract, multiply or divide by the same non-zero amount on both sides, the equation stays balanced and the solution does not change.
可以把方程想象成一架平衡的天平。如果你在两边同时加上、减去、乘以或除以同一个非零数，方程仍然保持平衡，解也不会改变。
The golden rule is: whatever you do to one side, you must also do to the other side.
黄金法则是：对等式一边所做的任何运算，必须同时对另一边做同样的运算。
3. Solving Equations with Addition and Subtraction | 含加法与减法的方程求解
If an equation has a number added to the unknown, subtract that number from both sides to isolate the variable.
如果方程中有数加在未知数上，应在等式两边同时减去这个数，以分离未知数。
Example: solve x + 7 = 15. Subtract 7 from both sides:
例如：解 x + 7 = 15。两边同时减去 7：
x + 7 − 7 = 15 − 7 ⇒ x = 8
If a number is subtracted from the unknown, add that number to both sides. For example, solve y − 3 = 9:
如果从未知数中减去了一个数，就在两边同时加上这个数。例如，解 y − 3 = 9：
y − 3 + 3 = 9 + 3 ⇒ y = 12
4. Solving Equations with Multiplication and Division | 含乘法与除法的方程求解
When the unknown is multiplied by a number, divide both sides by that number to find the value of the unknown.
当未知数乘以某个数时，两边同时除以这个数，即可求出未知数的值。
Example: solve 4x = 20. Divide both sides by 4:
例如：解 4x = 20。两边同时除以 4：
4x ÷ 4 = 20 ÷ 4 ⇒ x = 5
When the unknown is divided by a number, multiply both sides by that number. Example: solve x ÷ 5 = 3:
当未知数除以某个数时，两边同时乘以这个数。例如，解 x ÷ 5 = 3：
(x ÷ 5) × 5 = 3 × 5 ⇒ x = 15
5. Two-Step Equations | 两步方程
A two-step equation has two operations to undo. Use the reverse order of BIDMAS, so undo addition or subtraction first, then undo multiplication or division.
两步方程需要逆运算两个步骤。按照 BIDMAS 的逆顺序，先逆算加法或减法，再逆算乘法或除法。
Example: solve 2x + 3 = 11. First subtract 3 from both sides:
例如：解 2x + 3 = 11。首先两边同时减去 3：
2x + 3 − 3 = 11 − 3 ⇒ 2x = 8
Then divide both sides by 2:
然后两边同时除以 2：
2x ÷ 2 = 8 ÷ 2 ⇒ x = 4
Always use neat working and keep the equals signs in a straight line to avoid mistakes.
书写时一定要保持步骤整齐，等号对齐，这样可以减少错误。
6. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive law, then solve as normal.
当方程中含有括号时，先用乘法分配律展开括号，然后按正常步骤求解。
Example: solve 3(x + 2) = 18. Expand the bracket:
例如：解 3(x + 2) = 18。展开括号：
3x + 6 = 18
Subtract 6 from both sides:
两边同时减去 6：
3x = 12
Divide by 3:
再除以 3：
x = 4
y, you could divide both sides by 3 first, giving x + 2 = 6, then subtract 2 to get x = 4.
也可以先在两边同时除以 3，得到 x + 2 = 6，再减去 2，得到 x = 4。
7. Equations with Unknowns on Both Sides | 未知数在两边的情况
If an unknown appears on both sides, collect all unknown terms on one side and all number terms on the other side.
如果未知数出现在等式两边，应把含未知数的项移到一边，把常数项移到另一边。
Example: solve 5x + 2 = 3x + 10. Subtract 3x from both sides:
例如：解 5x + 2 = 3x + 10。两边同时减去 3x：
5x − 3x + 2 = 10 ⇒ 2x + 2 = 10
Subtract 2 from both sides:
两边同时减去 2：
2x = 8
Divide by 2:
再除以 2：
x = 4
It is often easier to keep the unknown positive by subtracting the smaller unknown term first.
通常先减去较小的未知数项，可以使未知数保持为正，计算起来更容易。
8. Equations with Fractional Coefficients | 含分数系数的方程
When the unknown has a fraction coefficient, multiply both sides by the denominator to clear the fraction.
当未知数的系数是分数时，两边同时乘以分母，以消去分数。
Example: solve 2x ÷ 3 = 6. Multiply both sides by 3:
例如：解 2x ÷ 3 = 6。两边同时乘以 3：
2x = 18
Divide by 2:
再除以 2：
x = 9
Example: solve x ÷ 4 + 1 = 3. First subtract 1:
例如：解 x ÷ 4 + 1 = 3。首先减去 1：
x ÷ 4 = 2
Then multiply both sides by 4:
然后两边同时乘以 4：
x = 8
9. Checking Your Solution | 检验答案
Always substitute your answer back into the original equation to check that the left-hand side equals the right-hand side.
解出答案后，一定要把答案代入原方程，检验等号左边是否等于右边。
Example: for 2x + 3 = 11, if x = 4:
例如：对于 2x + 3 = 11，若 x = 4：
LHS = 2(4) + 3 = 8 + 3 = 11 = RHS
This confirms that x = 4 is correct.
这可以确认 x = 4 是正确答案。
10. Common Mistakes to Avoid | 常见错误与规避
Many KS3 students lose marks by making small mechanical errors. The most common ones include:
许多 KS3 学生因为小的运算错误而失分。最常见的错误包括：
- Forgetting to do the same operation to both sides.
- 忘记对等式两边同时进行相同的运算。
- Incorrect signs when moving terms across the equals sign.
- 移项时符号出错。
- Undoing operations in the wrong order in two-step equations.
- 解两步方程时运算逆序错误。
- Dividing only part of the left-hand side instead of the entire side.
- 只除以左边的一部分，而不是整个左边。
To avoid these errors, write each step clearly and check the balance of the equation at every stage.
为了避免这些错误，每一步都要写清楚，并在每个阶段检查等式的平衡。
11. Real-Life Word Problems | 实际应用题
Linear equations are often used to solve word problems. The first step is to define the unknown with a letter, then build an equation from the given information.
一元一次方程常用于解应用题。第一步是用字母设未知数，然后根据已知信息建立方程。
Example: I think of a number, double it and add 5. The result is 17. Find the number.
例如：我想一个数，把它乘以 2 再加 5，结果是 17。求这个数。
Let the number be x:
设这个数为 x：
2x + 5 = 17
Subtract 5:
两边减 5：
2x = 12
Divide by 2:
再除以 2：
x = 6
So the original number is 6.
所以原来的数是 6。
12. Practice Quick Quiz | 快速练习
Try these questions to test your understanding. Solve each equation step by step, then check your answers.
试试下面的题目来检测你的理解。逐步解出每个方程，并检验答案。
| Equation | Solve for x |
|---|---|
| x + 9 = 20 | x = 11 |
| 5x = 35 | x = 7 |
| 3x − 4 = 14 | x = 6 |
| 2(x + 3) = 16 | x = 5 |
| 7x + 1 = 4x + 13 | x = 4 |
If you can solve these quickly and accurately, you have a solid foundation for linear equations at KS3 and beyond.
如果你能快速准确地解出这些方程，就说明你已经为 KS3 及更高阶段的一元一次方程打下了扎实的基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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