📚 Solving Linear Equations: Cambridge KS3 Maths (p317 Exercise Focus) | 剑桥 KS3 数学：解一元一次方程（p317 习题精讲）
Linear equations are the foundation of algebra at KS3. They show a balance between two expressions, and your job is to find the unknown value that makes the equation true. This revision guide walks you through the key methods, common errors, and exam-style examples you need for the Cambridge curriculum.
线性方程是 KS3 代数的基础。它们展示两个表达式之间的平衡关系，你的任务是找出使等式成立的未知数值。本复习指南将带你掌握剑桥课程所需的关键方法、常见错误和考试题型。
1. What Is a Linear Equation? | 什么是线性方程？
to the power of 1. It can be written in the form ax + b = c, where a, b and c are numbers.
线性方程是未知数（通常写作 x）只出现一次方且指数为 1 的方程。它可以写成 ax + b = c 的形式，其中 a、b、c 都是数。
2x + 3 = 11
Here, x is the unknown, 2 is the coefficient of x, 3 is the constant term, and 11 is the value the expression must equal.
这里，x 是未知数，2 是 x 的系数，3 是常数项，11 是表达式必须等于的值。
2. The Balance Method | 天平法
Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep it balanced. This is the golden rule of solving equations.
把方程想象成一个天平。无论你对一边做什么，都必须对另一边做同样的操作，才能保持平衡。这是解方程的黄金法则。
You can add, subtract, multiply or divide both sides by the same non-zero number without changing the solution.
你可以在两边同时加上、减去、乘以或除以同一个非零数，而不会改变方程的解。
| Operation | 操作 | Inverse Operation | 逆运算 |
|---|---|
| + 5 | − 5 |
| − 7 | + 7 |
| × 3 | ÷ 3 |
| ÷ 4 | × 4 |
3. Solving One-Step Equations | 解一步方程
One-step equations need only one inverse operation to isolate the unknown. Always write the operation you perform on both sides clearly.
一步方程只需要一次逆运算就能隔离未知数。始终要清楚地写出你在两边同时进行的运算。
Example 1: Solve x + 5 = 12.
例 1：解 x + 5 = 12。
x + 5 − 5 = 12 − 5 → x = 7
Example 2: Solve 3x = 18.
例 2：解 3x = 18。
3x ÷ 3 = 18 ÷ 3 → x = 6
Example 3: Solve x ÷ 4 = 5.
例 3：解 x ÷ 4 = 5。
x ÷ 4 × 4 = 5 × 4 → x = 20
4. Solving Two-Step Equations | 解两步方程
You must undo the addition or subtraction first, then undo the multiplication or division.
两步方程涉及未知数上的两步运算，例如 2x + 3 = 11。你必须先处理加法或减法，再处理乘法或除法。
Example: Solve 2x + 3 = 11.
例：解 2x + 3 = 11。
Step 1: Subtract 3 from both sides to remove the constant.
步骤 1：两边同时减 3，消去常数项。
2x + 3 − 3 = 11 − 3 → 2x = 8
Step 2: Divide both sides by 2 to isolate x.
步骤 2：两边同时除以 2，隔离 x。
2x ÷ 2 = 8 ÷ 2 → x = 4
Always work in the reverse order of operations: undo addition or subtraction before multiplication or division.
始终按照运算的逆顺序来操作：先处理加减，再处理乘除。
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive law. Then solve the resulting equation as usual.
当方程含有括号时，先用分配律展开括号，然后按常规方法解所得的方程。
Example: Solve 3(x + 2) = 21.
例：解 3(x + 2) = 21。
Step 1: Expand the bracket by multiplying each term inside by 3.
步骤 1：把括号内每一项都乘以 3，展开括号。
3x + 6 = 21
Step 2: Subtract 6 from both sides.
步骤 2：两边同时减 6。
3x + 6 − 6 = 21 − 6 → 3x = 15
Step 3: Divide both sides by 3.
步骤 3：两边同时除以 3。
x = 5
6. Equations with Unknowns on Both Sides | 未知数在方程两边的情况
If the unknown appears on both sides of the equation, collect all x terms on one side and all constant terms on the other side.
如果未知数出现在方程的两边，就把所有含 x 的项移到一边，把所有常数项移到另一边。
Example: Solve 5x − 2 = 2x + 10.
例：解 5x − 2 = 2x + 10。
Step 1: Subtract 2x from both sides so x only appears on the left.
步骤 1：两边同时减 2x，使 x 只出现在左边。
5x − 2x − 2 = 2x − 2x + 10 → 3x − 2 = 10
Step 2: Add 2 to both sides.
步骤 2：两边同时加 2。
3x − 2 + 2 = 10 + 2 → 3x = 12
Step 3: Divide both sides by 3.
步骤 3：两边同时除以 3。
x = 4
7. Equations with Fractional Coefficients | 分数系数的方程
When the unknown is divided by a number, multiply both sides by that denominator first. This clears the fraction and makes the equation easier to solve.
当未知数被一个数除时，先两边同时乘以这个分母。这样可以去分母，使方程更容易解。
Example: Solve x / 3 + 2 = 5.
例：解 x / 3 + 2 = 5。
Step 1: Subtract 2 from both sides.
步骤 1：两边同时减 2。
x / 3 + 2 − 2 = 5 − 2 → x / 3 = 3
Step 2: Multiply both sides by 3.
步骤 2：两边同时乘以 3。
x / 3 × 3 = 3 × 3 → x = 9
For equations like 2x / 5 = 6, multiply both sides by 5 first, then divide by 2.
对于像 2x / 5 = 6 这样的方程，先两边同时乘以 5，再除以 2。
2x / 5 × 5 = 6 × 5 → 2x = 30 → x = 15
8. Word Problems Leading to Linear Equations | 列方程解应用题
Word problems ask you to translate everyday language into an equation. Let the unknown be x, write down the relationship, then solve.
应用题要求你把日常语言翻译成方程。设未知数为 x，写出关系式，然后求解。
Example: Three times a number plus 5 equals 20. Find the number.
例：一个数的 3 倍加 5 等于 20。求这个数。
Let the number be x. The equation is 3x + 5 = 20.
设这个数为 x。方程为 3x + 5 = 20。
3x + 5 = 20 → 3x = 15 → x = 5
Always check the wording: 'more than' means addition, 'less than' means subtraction, 'times' means multiplication, and 'divided by' means division.
始终注意措辞：'more than' 表示加，'less than' 表示减，'times' 表示乘，'divided by' 表示除。
9. Checking Your Answers | 验证答案
Always substitute your answer back into the original equation to make sure both sides are equal. This takes only a few seconds and can catch small mistakes.
始终把你的答案代回原方程，确认两边相等。这只需几秒钟，却能发现细小的错误。
Example: For 2x + 3 = 11, we found x = 4. Substitute x = 4.
例：对于 2x + 3 = 11，我们解得 x = 4。代入 x = 4。
2 × 4 + 3 = 8 + 3 = 11 ✓
The two sides match, so the solution is correct.
两边相等，所以解是正确的。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many marks are lost not because the method is hard, but because of small slips. Watch out for these common errors.
很多失分并不是因为方法难，而是因为小的疏忽。请注意以下常见错误。
- Forgetting to do the same thing to both sides. If you subtract 3 on the left, you must subtract 3 on the right too.
- 忘记在两边同时进行相同操作。如果你在左边减 3，右边也必须减 3。
- part of the expression. In 2x + 3 = 11, you must subtract 3 first, not divide by 2 first.
- 只除以表达式的一部分。在 2x + 3 = 11 中，你必须先减 3，而不是先除以 2。
- Making a sign error when moving terms across the equal sign.
- 移项时出现符号错误。
- Forgetting to expand brackets completely before solving.
- 在解方程前忘记完全展开括号。
Always show each step clearly. In Cambridge KS3 exams, method marks can be awarded even if the final answer is wrong.
始终清晰地写出每一步。在剑桥 KS3 考试中，即使最终答案错误，也可能得到方法分。
11. Practice Questions | 练习题
Try these exam-style questions. Solve each equation and check your answer.
尝试以下考试题型。解每个方程并检查你的答案。
- 4x + 7 = 31
- x / 2 − 3 = 6
- 5(x − 2) = 25
- 7x + 4 = 3x + 20
Answers: x = 6, x = 18, x = 7, x = 4.
答案：x = 6，x = 18，x = 7，x = 4。
12. Summary | 总结
To solve any KS3 linear equation, keep the balance, undo operations in reverse order, expand brackets, collect like terms, and check your solution. With practice, these steps become automatic.
要解任何 KS3 线性方程，就要保持平衡、按逆顺序进行运算、展开括号、合并同类项，并检验你的解。通过练习，这些步骤会变得自然而然。
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