📚 Solving Linear Equations | 解一元一次方程
Linear equations are the foundation of algebra at KS3. A linear equation is an equation where the unknown, usually written as x, is raised only to the power of 1. Mastering how to solve these equations helps students tackle more advanced topics in the Cambridge Lower Secondary Mathematics curriculum.
线性方程是 KS3 代数的基础。一元一次方程是未知数（通常写为 x）的指数只为 1 的方程。掌握解这类方程的方法，有助于学生应对 Cambridge 初中数学课程中更高级的课题。
1. What is a Linear Equation? | 什么是一元一次方程
A linear equation has the general form ax + b = c, where a, b and c are numbers and a is not zero. The unknown x appears only to the power 1. This means there are no x², x³ or 1/x terms.
一元一次方程的一般形式为 ax + b = c，其中 a、b、c 是数，且 a 不等于零。未知数 x 只以指数 1 出现，也就是说没有 x²、x³ 或 1/x 这样的项。
For example, 2x + 3 = 11 is a linear equation. The solution is the value of x that makes the equation true.
例如，2x + 3 = 11 是一元一次方程。方程的解就是使等式成立的 x 的值。
- 2x + 3 = 11 is linear - it has x to the power 1 only. | 2x + 3 = 11 是线性的 - 它只含有 x 的 1 次方。
- x² + 5 = 9 is not linear - it has x to the power 2. | x² + 5 = 9 不是线性的 - 它含有 x 的 2 次方。
- 1/x = 4 is not linear - x is in the denominator. | 1/x = 4 不是线性的 - x 在分母上。
Linear equations are the simplest type of equation to solve, but they appear in many areas of mathematics and science.
一元一次方程是最容易求解的方程类型，但在数学和科学的许多领域都会出现。
2. The Balance Method | 天平法
To solve a linear equation, we use the balance method. Think of an equation as a set of balance scales. The left side and the right side must always stay equal.
解一元一次方程时，我们使用天平法。可以把方程想象成一台天平，左边和右边必须始终保持平衡。
Whatever operation you do to one side of the equation, you must do exactly the same to the other side. This keeps the equation balanced.
无论对方程的一边做了什么运算，都必须对另一边做完全相同的运算。这样方程才能保持平衡。
x + 3 = 7
x + 3 − 3 = 7 − 3
Here we subtract 3 from both sides to isolate x. The result is x = 4.
这里我们在两边同时减去 3，从而把 x 单独留在一边。结果是 x = 4。
The balance method is the key skill behind every correct solution. If you forget to apply an operation to both sides, you will get the wrong answer.
天平法是每一个正确解答背后的关键技能。如果忘记在两边同时进行运算，就会得到错误答案。
3. Solving by Addition and Subtraction | 通过加法与减法求解
When an equation has a number added to or subtracted from x, we use the inverse operation to remove that number. Addition and subtraction are inverse operations.
当方程中有某个数加上或减去 x 时，我们使用逆运算来去掉这个数。加法和减法互为逆运算。
For example, solve x + 5 = 12. Since 5 is added to x, we subtract 5 from both sides.
例如，解 x + 5 = 12。因为 5 加在 x 上，所以我们从两边同时减去 5。
x + 5 − 5 = 12 − 5
x = 7
Now solve x − 4 = 9. Since 4 is subtracted from x, we add 4 to both sides.
现在解 x − 4 = 9。因为 4 从 x 中减去，所以我们在两边同时加上 4。
x − 4 + 4 = 9 + 4
x = 13
Always write the operation you are applying on both sides. This shows your working clearly and helps you avoid mistakes.
一定要把两边同时进行的运算写出来。这样能清晰地展示解题过程，并帮助你避免错误。
4. Solving by Multiplication and Division | 通过乘法与除法求解
When x is multiplied by a number or divided by a number, we use the inverse operation to isolate x. Multiplication and division are inverse operations.
当 x 乘以或除以某个数时，我们使用逆运算来分离出 x。乘法和除法互为逆运算。
For example, solve 3x = 18. The 3 is multiplied by x, so we divide both sides by 3.
例如，解 3x = 18。3 与 x 相乘，所以我们将两边同时除以 3。
3x ÷ 3 = 18 ÷ 3
x = 6
Now solve x/4 = 5. The x is divided by 4, so we multiply both sides by 4.
现在解 x/4 = 5。x 除以 4，所以我们将两边同时乘以 4。
(x/4) × 4 = 5 × 4
x = 20
These one-step operations are the building blocks for more complicated equations. Practise them until they feel automatic.
这些一步运算是更复杂方程的基础。要多练习，直到能够自动完成。
5. Two-Step Equations | 两步方程
A two-step equation has two operations applied to x. To solve it, we reverse the order of operations. Usually we undo addition or subtraction first, then multiplication or division.
两步方程对 x 进行了两种运算。解这类方程时，我们要反向进行运算顺序。通常先去加法或减法，再去乘法或除法。
Solve 2x + 3 = 11. First subtract 3 from both sides, then divide both sides by 2.
解 2x + 3 = 11。首先在两边同时减去 3，然后在两边同时除以 2。
2x + 3 − 3 = 11 − 3
2x = 8
2x ÷ 2 = 8 ÷ 2
x = 4
Another example is 5x − 2 = 23. Add 2 to both sides first, then divide by 5.
另一个例子是 5x − 2 = 23。先在两边同时加上 2，再同时除以 5。
5x − 2 + 2 = 23 + 2
5x = 25
5x ÷ 5 = 25 ÷ 5
x = 5
Remember: the order you undo operations is the reverse of the order they were applied to x. This matches the idea of undoing the last step first.
记住：你解除运算的顺序应该与这些运算作用于 x 的顺序相反。这符合“先撤销最后一步”的思路。
6. Equations with Brackets | 含括号的方程
Some linear equations include brackets, such as 3(x + 2) = 21. There are two common ways to solve them: divide first or expand first.
有些一元一次方程含有括号，例如 3(x + 2) = 21。解这类方程有两种常用方法：先除以系数，或先展开括号。
Method 1: Divide both sides by 3 first. This removes the coefficient in front of the bracket.
方法一：先在两边同时除以 3。这样可以去掉括号前面的系数。
3(x + 2) ÷ 3 = 21 ÷ 3
x + 2 = 7
x + 2 − 2 = 7 − 2
x = 5
Method 2: Expand the bracket first, then solve as a two-step equation.
方法二：先展开括号，然后把它当作两步方程来解。
3(x + 2) = 21
3x + 6 = 21
3x + 6 − 6 = 21 − 6
3x = 15
3x ÷ 3 = 15 ÷ 3
x = 5
Both methods give the same answer. Dividing first often saves time, but expanding first is sometimes easier when the division is not exact.
两种方法都能得到相同的答案。先除以系数通常更省时间，但如果除法不能整除，先展开括号有时更容易。
7. Equations with Unknowns on Both Sides | 未知数在两侧的方程
Some equations have the unknown x on both sides, for example 5x + 2 = 3x + 10. The key idea is to collect all x terms on one side and all number terms on the other side.
有些方程的两边都含有未知数 x，例如 5x + 2 = 3x + 10。关键思路是把所有含 x 的项移到一边，把所有数字项移到另一边。
First subtract 3x from both sides so that the x terms are only on the left.
首先在两边同时减去 3x，使含 x 的项只出现在左边。
5x + 2 − 3x = 3x + 10 − 3x
2x + 2 = 10
Now solve the two-step equation by subtracting 2 from both sides, then dividing by 2.
现在解这个两步方程：先在两边同时减去 2，再同时除以 2。
2x + 2 − 2 = 10 − 2
2x = 8
2x ÷ 2 = 8 ÷ 2
x = 4
You could also collect the x terms on the right instead. As long as you apply the same operation to both sides, you will get the same solution.
你也可以把含 x 的项集中到右边。只要在两边进行相同的运算，就能得到相同的解。
8. Word Problems Leading to Equations | 由文字题建立方程
Many exam questions describe a situation in words. You need to translate the words into an equation using x as the unknown number.
许多考试题目会用文字描述一个情境。你需要把文字翻译成方程，用 x 表示未知数。
Example: 'I think of a number, double it and add 5. The result is 19. Find the number.'
例如：“我想一个数，把它乘以 2，再加上 5，结果是 19。求这个数。”
Let the number be x. Double it means 2x. Adding 5 gives 2x + 5. The result is 19, so:
设这个数为 x。把它乘以 2 就是 2x。再加上 5 得到 2x + 5。结果是 19，所以：
2x + 5 = 19
Now solve it by subtracting 5 from both sides and then dividing by 2.
现在解这个方程：先在两边同时减去 5，再同时除以 2。
2x + 5 − 5 = 19 − 5
2x = 14
2x ÷ 2 = 14 ÷ 2
x = 7
The number is 7. Always write the final answer in words when the question is a word problem.
这个数是 7。对于文字题，最终答案一定要用文字写出来。
9. Checking Your Solution | 检验你的解
After you find a solution, substitute it back into the original equation to check that both sides are equal. This is called checking by substitution.
求出解之后，把它代回原方程，检查两边是否相等。这叫做代入检验。
For 2x + 3 = 11, we found x = 4. Substitute x = 4 into the left side:
对于 2x + 3 = 11，我们得到 x = 4。把 x = 4 代入左边：
2(4) + 3 = 8 + 3 = 11
The left side equals the right side, so x = 4 is correct.
左边等于右边，所以 x = 4 是正确的。
Checking is especially important in exams because it helps you catch small arithmetic mistakes before you move on.
检验在考试中尤其重要，因为它能在你继续答题之前帮你发现小的计算错误。
10. Common Mistakes to Avoid | 常见错误与避免方法
Many students lose marks on linear equations because of avoidable mistakes. The most common is forgetting to apply the same operation to both sides.
许多学生在一元一次方程上失分，是因为一些本可避免的错误。最常见的错误是忘记在方程两边同时进行相同的运算。
- Forgetting to subtract from both sides: 'x + 5 = 12 so x = 17' is wrong because 5 was added to the right side instead. | 忘记在两边同时减去同一个数：“x + 5 = 12 所以 x = 17” 是错误的，因为右边实际上多加了 5。
- Only dividing part of one side: '2x + 3 = 11 so x + 3 = 5.5' is wrong because only part of the left side was divided. | 只除以一边的部分项：“2x + 3 = 11 所以 x + 3 = 5.5” 是错误的，因为只对左边的部分项进行了除法。
- Sign errors when moving terms: 'x − 4 = 9 so x = 5' is wrong because the −4 was not changed to +4. | 移项时符号错误：“x − 4 = 9 所以 x = 5” 是错误的，因为 −4 没有变为 +4。
- Forgetting to check the final answer. | 忘记检验最终答案。
To avoid these mistakes, write down every step clearly and always use the balance method.
为了避免这些错误，要把每一步都写清楚，并且始终使用天平法。
11. Practice Questions | 练习题
Try these practice questions. Solve each equation and then check your answer by substitution.
试着做以下练习题。解出每个方程，然后用代入法检验答案。
- x + 7 = 15 | x + 7 = 15
- x − 3 = 8 | x − 3 = 8
- 4x = 32 | 4x = 32
- x/5 = 6 | x/5 = 6
- 3x + 2 = 17 | 3x + 2 = 17
- 5x − 4 = 21 | 5x − 4 = 21
- 2(x + 3) = 16 | 2(x + 3) = 16
- 4x + 1 = 2x + 9 | 4x + 1 = 2x + 9
The answers are: x = 8, x = 11, x = 8, x = 30, x = 5, x = 5, x = 5, x = 4.
答案分别是：x = 8，x = 11，x = 8，x = 30，x = 5，x = 5，x = 5，x = 4。
Work slowly and show every step. Marks are often awarded for correct working even if the final answer is wrong.
慢慢做，写出每一步。即使最终答案有误，正确的解题过程通常也能得分。
12. Summary and Exam Tips | 总结与考试技巧
Solving a linear equation means finding the value of x that makes the equation true. Use the balance method at every step.
解一元一次方程就是求出使等式成立的 x 的值。每一步都要使用天平法。
Key steps: undo addition or subtraction first, then undo multiplication or division. If there are brackets, either divide first or expand first. If unknown terms appear on both sides, collect them on one side.
关键步骤：先去加法或减法，再去乘法或除法。如果有括号，可以先除以系数，也可以先展开。如果未知数在两边都出现，要把它们移到同一边。
Always check your solution by substituting it back into the original equation. In word problems, write the final answer clearly in words.
一定要把解代回原方程进行检验。对于文字题，最终答案要用文字清楚地写出。
Practise regularly with small numbers and then with negative numbers and fractions. This builds both speed and accuracy for Cambridge KS3 assessments.
要坚持用小数、负数和分数进行练习。这能提升你在 Cambridge KS3 考试中的速度和准确率。
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