📚 Solving Linear Equations: The Balance Method | 解一元一次方程：天平法
Linear equations are one of the most important building blocks in KS3 mathematics, especially in the Cambridge Checkpoint curriculum. They help you model real-life problems using algebra and prepare you for more advanced topics such as simultaneous equations and functions.
一元一次方程是 KS3 数学中最重要的基础内容之一，尤其在剑桥 Checkpoint 课程中。它们帮助你用代数建模现实生活中的问题，并为后续更高级的主题（如联立方程和函数）做好准备。
In this article, you will learn how to solve linear equations step by step using the balance method. We will start with simple one-step equations and progress to equations with brackets and unknowns on both sides. Each section includes worked examples and common mistakes to avoid.
在本文中，你将学习如何使用天平法一步一步地解一元一次方程。我们将从简单的一步方程开始，逐步深入到含括号和未知数在等号两侧的方程。每一节都包含解题示例和需要避免的常见错误。
1. What is a Linear Equation? | 什么是一元一次方程？
A linear equation is an equation in which the highest power of the unknown variable is 1. In KS3, the unknown is usually written as x, but it can also be any letter such as y, a, or n.
一元一次方程是指未知数的最高次数为 1 的方程。在 KS3 阶段，未知数通常写成 x，但也可以是任何字母，例如 y、a 或 n。
The general form of a simple linear equation is:
简单一元一次方程的一般形式为：
ax + b = c
Here, a, b, and c are numbers, and x is the unknown we need to find. For example, in the equation 2x + 3 = 7, the coefficient of x is 2, the constant added is 3, and the right-hand side is 7.
这里，a、b 和 c 是数字，x 是我们要找的未知数。例如，在方程 2x + 3 = 7 中，x 的系数是 2，加上的常数是 3，右边是 7。
Solving a linear equation means finding the value of x that makes the equation true. In the example above, the solution is x = 2 because 2 × 2 + 3 = 7.
解一元一次方程意味着求出使方程成立的 x 值。在上面的例子中，解是 x = 2，因为 2 × 2 + 3 = 7。
2. The Balance Method | 天平法
Think of an equation as a balance scale. The equals sign tells you that the left-hand side and the right-hand side have exactly the same value. If you add, subtract, multiply, or divide one side by a number, you must do the same to the other side to keep the scale balanced.
把方程想象成一台天平。等号告诉你左边和右边的值完全相同。如果你对一边加上、减去、乘以或除以一个数，你必须对另一边做同样的操作，才能保持天平平衡。
This rule is the foundation of all algebraic equation solving. It is often called the balancing rule or the golden rule of equations.
这条规则是解所有代数方程的基础。它通常被称为平衡规则或方程的金科玉律。
For example, if x + 3 = 7, you can subtract 3 from both sides to keep the balance:
例如，如果 x + 3 = 7，你可以在两边同时减去 3 以保持平衡：
x + 3 − 3 = 7 − 3
This simplifies to x = 4. As long as you perform the same operation on both sides, the equation remains true.
化简后得到 x = 4。只要你对两边执行相同的操作，方程就仍然成立。
3. Solving by Adding or Subtracting | 通过加减解方程
When the unknown has a number added to it or subtracted from it, use the inverse operation to isolate x. The inverse of addition is subtraction, and the inverse of subtraction is addition.
当未知数被加上或减去一个数时，使用逆运算来分离 x。加法的逆运算是减法，减法的逆运算是加法。
Worked example 1: Solve x + 5 = 12.
解题示例 1：解方程 x + 5 = 12。
x + 5 − 5 = 12 − 5
x = 7
Worked example 2: Solve x − 9 = 4.
解题示例 2：解方程 x − 9 = 4。
x − 9 + 9 = 4 + 9
x = 13
Always write the operation you are doing on both sides. This helps you avoid mistakes and makes your working clear to examiners.
始终把你正在做的运算写在两边。这有助于避免错误，也能让阅卷人清楚地看到你的解题过程。
4. Solving by Multiplying or Dividing | 通过乘除解方程
If x is multiplied by a number, divide both sides by that number. If x is divided by a number, multiply both sides by that number.
如果 x 乘以了一个数，就在两边同时除以这个数。如果 x 除以了一个数，就在两边同时乘以这个数。
Worked example 3: Solve 3x = 18.
解题示例 3：解方程 3x = 18。
3x ÷ 3 = 18 ÷ 3
x = 6
Worked example 4: Solve x ÷ 4 = 5.
解题示例 4：解方程 x ÷ 4 = 5。
x ÷ 4 × 4 = 5 × 4
x = 20
Remember that division and multiplication by a non-zero number do not change the equality, as long as you apply the operation to both sides.
请记住，只要你在两边进行乘或除以一个非零数的操作，等式的平衡就不会改变。
5. Two-Step Equations | 两步方程
A two-step equation has two operations attached to x, such as multiplication and addition. To solve it, undo the operations in reverse order: first deal with addition or subtraction, then deal with multiplication or division.
两步方程是指 x 上有两种运算，例如乘法和加法。解这类方程时，要按相反顺序消去运算：先处理加法或减法，再处理乘法或除法。
Worked example 5: Solve 2x + 3 = 11.
解题示例 5：解方程 2x + 3 = 11。
2x + 3 − 3 = 11 − 3
2x = 8
2x ÷ 2 = 8 ÷ 2
x = 4
Worked example 6: Solve 5x − 7 = 13.
解题示例 6：解方程 5x − 7 = 13。
5x − 7 + 7 = 13 + 7
5x = 20
5x ÷ 5 = 20 ÷ 5
x = 4
Many KS3 exam questions ask you to show all steps. Writing each line below the previous one is an excellent habit.
许多 KS3 考试题要求你写出所有步骤。把每一步写在前一步的下一行是一个很好的习惯。
6. Equations with Brackets | 含括号的方程
Equations often contain brackets such as 2(x + 4) = 14. You can solve them by expanding the bracket first or by dividing both sides by the number outside the bracket.
方程中经常带有括号，例如 2(x + 4) = 14。你可以先展开括号，也可以在两边同时除以括号外的数。
Method A: Expand the bracket first.
方法 A：先展开括号。
2(x + 4) = 14
2x + 8 = 14
2x + 8 − 8 = 14 − 8
2x = 6
x = 3
Method B: Divide both sides by 2 first.
方法 B：先在两边同时除以 2。
2(x + 4) ÷ 2 = 14 ÷ 2
x + 4 = 7
x = 3
Both methods give the same answer. Choose the method that feels easier for the numbers involved.
两种方法得到相同的答案。选择对题中数字更简便的方法即可。
7. Unknowns on Both Sides | 未知数在两侧的方程
Sometimes the unknown appears on both sides of the equals sign, for example 5x + 2 = 2x + 11. To solve, collect all x terms on one side and all number terms on the other side.
有时未知数出现在等号的两侧，例如 5x + 2 = 2x + 11。解这类方程时，要把所有含 x 的项移到一边，把所有数字项移到另一边。
Worked example 7: Solve 5x + 2 = 2x + 11.
解题示例 7：解方程 5x + 2 = 2x + 11。
5x − 2x + 2 = 2x − 2x + 11
3x + 2 = 11
3x + 2 − 2 = 11 − 2
3x = 9
x = 3
Worked example 8: Solve 7x − 4 = 3x + 8.
解题示例 8：解方程 7x − 4 = 3x + 8。
7x − 3x − 4 = 3x − 3x + 8
4x − 4 = 8
4x = 12
x = 3
Always be careful with negative signs when moving terms. A common mistake is to change the sign incorrectly when taking a term across the equals sign.
移动项时要特别注意负号。一个常见错误是把某一项移到等号另一边时，符号变错了。
8. Checking Your Solution | 检验你的解
After you find a solution, substitute it back into the original equation to check that the left-hand side equals the right-hand side. This takes only a few seconds and can help you catch mistakes.
求出解后，把它代回原方程，检查左边是否等于右边。这只需几秒钟，却可以帮助你发现错误。
For example, if you solved 2x + 3 = 9 and got x = 3, check:
例如，如果你解方程 2x + 3 = 9 得到 x = 3，检验如下：
2(3) + 3 = 6 + 3 = 9
Since the left-hand side equals the right-hand side, the solution is correct. If the two sides do not match, re-check your working line by line.
由于左边等于右边，解是正确的。如果两边不相等，请逐行重新检查你的解题过程。
9. Common Mistakes to Avoid | 常见错误及避免方法
Here are some frequent errors students make when solving linear equations. Knowing these will help you avoid losing marks.
以下是学生在解一元一次方程时常见的一些错误。了解这些错误可以帮助你避免丢分。
Forgetting to do the same operation on both sides. For example, writing x + 3 = 7 and then saying x = 7 without subtracting 3.
忘记在两边执行相同的运算。例如，写下 x + 3 = 7 后，不减 3 就说 x = 7。
Expanding brackets incorrectly, such as writing 2(x + 4) = 2x + 4 instead of 2x + 8.
去括号时出错，例如把 2(x + 4) 写成 2x + 4，而不是 2x + 8。
Changing the sign incorrectly when moving terms across the equals sign.
把项移到等号另一边时符号变错。
Dividing by the coefficient but forgetting that the coefficient is negative, for example solving −2x = 10 and writing x = 10 instead of x = −5.
除以系数时忘记系数为负，例如解 −2x = 10 时写成 x = 10，而正确应为 x = −5。
To avoid these errors, always write down each operation on both sides and check your final answer by substitution.
为了避免这些错误，请始终写出两边进行的每一步运算，并通过代入检验最终答案。
10. Practice Questions and Worked Examples | 练习题与解答示例
Try these practice questions. Cover the answers first, then check your work step by step.
尝试以下练习题。先盖住答案，然后逐步检查你的解题过程。
| Question | 题目 | Answer | 答案 |
|---|---|
| x + 9 = 21 | x = 12 |
| 4x = 32 | x = 8 |
| 3x + 2 = 17 | x = 5 |
| 5(x − 2) = 20 | x = 6 |
| 6x + 1 = 2x + 17 | x = 4 |
Worked solution for 3x + 2 = 17:
题目 3x + 2 = 17 的详细解答如下：
3x + 2 − 2 = 17 − 2
3x = 15
x = 5
Worked solution for 5(x − 2) = 20:
题目 5(x − 2) = 20 的详细解答如下：
5(x − 2) ÷ 5 = 20 ÷ 5
x − 2 = 4
x = 6
Practising these types of questions regularly will build your confidence and speed. Make sure you can solve them without looking at the answers.
定期练习这些类型的题目可以增强你的信心和解题速度。确保你能在不看答案的情况下独立完成它们。
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