📚 Solving Linear Equations | 求解线性方程
Linear equations are the foundation of A-level algebra. Before studying quadratics, simultaneous equations, or functions, you need to be completely fluent in rearranging and solving equations that involve a variable raised only to the first power.
线性方程是 A-level 代数的基础。在学习二次方程、联立方程或函数之前，你必须非常熟练地整理并求解仅含一次幂变量的方程。
1. What Is a Linear Equation? | 什么是线性方程？
A linear equation in one unknown is any equation that can be written in the form ax + b = 0, where a and b are constants and a ≠ 0.
一元线性方程是任何可以写成 ax + b = 0 形式的方程，其中 a 和 b 为常数，且 a ≠ 0。
In AQA papers, the unknown is usually labelled x, but it can also appear as y, t, n, or another letter. The key idea is that no powers such as x², x³, or 1/x occur.
在 AQA 试题中，未知数通常用 x 表示，但也可能是 y、t、n 或其他字母。关键点在于不会出现 x²、x³ 或 1/x 等幂次。
2. The Balance Method and Inverse Operations | 天平法与逆运算
Think of an equation as a balance scale. Whatever you do to one side, you must also do to the other side to keep the equation true.
可以把方程想象成一个天平。无论你对一边做什么运算，对另一边也必须做同样的运算，方程才能保持成立。
To isolate the unknown, apply inverse operations in the reverse order of the original operations. Addition and subtraction are inverse operations; multiplication and division are also inverse operations.
为了分离未知数，需要按原运算的相反顺序应用逆运算。加法与减法互为逆运算；乘法与除法也互为逆运算。
x + 5 = 12 ⇒ x = 12 − 5 = 7
For the equation 2x = 14, multiplication by 2 is reversed by dividing both sides by 2, giving x = 7.
对于方程 2x = 14，乘以 2 的逆运算是两边同时除以 2，得到 x = 7。
3. Solving One-Step Equations | 解一步方程
A one-step equation requires only one inverse operation. For addition or subtraction, subtract or add the same number on both sides.
一步方程只需一步逆运算即可求解。对于加法或减法形式，在两边同时减去或加上同一个数。
- English: Solve x + 6 = 15. Chinese: 求解 x + 6 = 15。 Subtract 6 from both sides: x = 9. 两边同时减去 6：x = 9。
- English: Solve x − 4 = 10. Chinese: 求解 x − 4 = 10。 Add 4 to both sides: x = 14. 两边同时加上 4：x = 14。
For multiplication or division, divide or multiply both sides by the same non-zero number.
对于乘法或除法形式，在两边同时除以或乘以同一个非零数。
5x = 35 ⇒ x = 35 ÷ 5 = 7
x ÷ 3 = 8 ⇒ x = 8 × 3 = 24
4. Solving Two-Step Equations | 解两步方程
Two-step equations involve two operations on the unknown, such as multiplication followed by addition. Always reverse the addition or subtraction first, then reverse the multiplication or division.
两步方程含对未知数进行两次运算，例如先乘后加。求解时通常先逆算加减法，再逆算乘除法。
For example, to solve 2x + 3 = 11, subtract 3 from both sides first, then divide by 2.
例如，解 2x + 3 = 11，先两边同时减去 3，再同时除以 2。
2x + 3 = 11 ⇒ 2x = 8 ⇒ x = 4
Similarly, for x/5 − 2 = 3, add 2 to both sides, then multiply both sides by 5 to obtain x = 25.
类似地，对于 x/5 − 2 = 3，两边先加 2，再同时乘以 5，得到 x = 25。
5. Expanding Brackets | 展开括号
If an equation contains brackets, expand them first using the distributive law. For example, 3(x + 2) = 21 becomes 3x + 6 = 21.
如果方程含有括号，应先用分配律展开。例如 3(x + 2) = 21 展开为 3x + 6 = 21。
3(x + 2) = 21 ⇒ 3x + 6 = 21 ⇒ 3x = 15 ⇒ x = 5
Be careful when a negative sign is outside the bracket. In 4 − 2(x − 3) = 10, expand −2(x − 3) as −2x + 6, not −2x − 6.
括号外有负号时要特别小心。在 4 − 2(x − 3) = 10 中，−2(x − 3) 应展开为 −2x + 6，而不是 −2x − 6。
6. Unknowns on Both Sides | 未知数在方程两边
When the unknown appears on both sides of an equation, collect all variable terms on one side and constant terms on the other.
当未知数出现在方程两边时，应把含未知数的项移到一边，常数项移到另一边。
For 5x + 2 = 3x + 10, subtract 3x from both sides to get 2x + 2 = 10, then subtract 2 and divide by 2 to get x = 4.
对于 5x + 2 = 3x + 10，两边同时减去 3x 得到 2x + 2 = 10，再减去 2 并除以 2，得到 x = 4。
5x + 2 = 3x + 10 ⇒ 2x = 8 ⇒ x = 4
You can choose to collect the variable term on either side. It is usually easier to aim for a positive coefficient of the unknown.
你可以选择把变量项移到任意一边。通常使未知数的系数为正会更容易计算。
7. Equations with Fractions | 含分数的方程
To solve equations involving fractions, multiply every term by the lowest common denominator to clear the fractions.
解含有分数的方程时，可将每一项都乘以最小公分母，从而消去分母。
For example, in (x + 1)/3 = 4, multiply both sides by 3 to get x + 1 = 12, so x = 11.
例如，在 (x + 1)/3 = 4 中，两边乘以 3 得到 x + 1 = 12，因此 x = 11。
For (2x)/5 + 1 = 3, first subtract 1 to get (2x)/5 = 2, then multiply by 5 and divide by 2 to find x = 5.
对于 (2x)/5 + 1 = 3，先减去 1 得到 (2x)/5 = 2，再乘以 5 并除以 2，得到 x = 5。
If two fractions are equal, cross multiplication can be used: a/b = c/d ⇒ ad = bc, provided no denominator is zero.
如果两个分式相等，可以使用交叉相乘：a/b = c/d ⇒ ad = bc，前提是分母不为零。
8. Special Cases in Linear Equations | 线性方程的特殊情况
Some linear equations are identities: they are true for every value of the unknown. For example, 2(x + 3) = 2x + 6 simplifies to 2x + 6 = 2x + 6.
有些线性方程是恒等式：对未知数的任何取值都成立。例如 2(x + 3) = 2x + 6 化简后为 2x + 6 = 2x + 6。
Other equations are contradictions: they are never true. For example, 2x + 3 = 2x + 5 gives 3 = 5, which is impossible, so there is no solution.
另一些方程是矛盾方程：它们永远不成立。例如 2x + 3 = 2x + 5 化简后得到 3 = 5，这不可能成立，因此无解。
In an exam, if all variable terms cancel, stop and decide whether the resulting statement is always true or always false.
考试中如果所有含未知数的项都相消了，先停下来判断剩下的等式是恒真还是恒假。
9. Setting Up Linear Equations from Word Problems | 根据文字题建立线性方程
AQA questions often ask you to form an equation from a real-life situation before solving it. Define the unknown clearly, then translate the words into algebra.
AQA 试题经常要求你先根据实际情况建立方程，再求解。要清楚地设出未知数，然后把文字转化为代数式。
Example: The perimeter of a rectangle is 34 cm. Its length is 2 cm longer than twice its width. Let the width be x cm. Then the length is 2x + 2 cm.
例题：一个矩形的周长为 34 cm，其长比宽的 2 倍多 2 cm。设宽为 x cm，则长为 2x + 2 cm。
2(x + 2x + 2) = 34 ⇒ 2(3x + 2) = 34 ⇒ 6x + 4 = 34 ⇒ x = 5
Always check that your answer makes sense in the original context, not just in the algebra.
务必检查答案在原情境中是否合理，而不仅仅是在代数式中成立。
10. Checking Solutions and Common Mistakes | 验证解与常见错误
Substitute your final value back into the original equation to confirm that both sides are equal.
把最终得到的值代回原方程，确认两边是否相等。
Common mistakes include sign errors when expanding brackets, dividing before subtracting, and multiplying only part of an equation by a denominator.
常见错误包括展开括号时符号出错、先除后减的顺序错误，以及乘以分母时只乘了方程的一部分。
| Common mistake | 常见错误 | Correct approach | 正确做法 |
|---|---|
| 2(x + 3) = 2x + 3 | 2(x + 3) = 2x + 6 |
| x/3 + 1 = 5 ⇒ x + 1 = 5 | x/3 + 1 = 5 ⇒ x/3 = 4 ⇒ x = 12 |
| 3x − 2 = 7 ⇒ 3x = 5 | 3x − 2 = 7 ⇒ 3x = 9 |
11. AQA Exam-Style Strategy | AQA 考试风格策略
In AQA examinations, method marks are awarded for correct algebraic steps even if the final answer is wrong. Always show each stage of working clearly.
在 AQA 考试中，即使最终答案错误，正确的代数步骤也能得分。因此要清晰写出每一步推导过程。
Keep the equation balanced and write one operation per line. Avoid trying to do too many steps mentally, especially when negative signs or fractions are involved.
保持方程平衡，每行只写一步运算。当涉及负号或分数时，尽量避免过多心算。
If a question asks for an exact answer, give it in exact form. For linear equations, the exact answer is usually a rational number, such as a fraction or an integer.
如果题目要求精确答案，请给出精确形式。对于线性方程，精确答案通常是有理数，如分数或整数。
12. Summary | 总结
To solve a linear equation confidently, use the balance method, apply inverse operations, expand brackets correctly, and clear fractions early.
要熟练求解线性方程，可运用天平法、应用逆运算、正确展开括号，并尽早消去分母。
- English: Simplify both sides first. Chinese: 先化简两边。
- English: Collect variable terms on one side. Chinese: 把含未知数的项移到一边。
- English: Isolate the variable by inverse operations. Chinese: 通过逆运算分离未知数。
- English: Check by substitution. Chinese: 代入原方程检验。
Mastering linear equations gives you the algebraic fluency needed for higher A-level topics such as functions, graphs, and modelling.
掌握线性方程能为你学习函数、图像和数学建模等更高阶 A-level 内容打下代数基础。
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