📚 Solving Linear Equations | 解一元一次方程
Linear equations are the building blocks of algebra at KS3. They appear in many different forms, and being able to solve them confidently is essential for success in the Cambridge Checkpoint and beyond. This article covers the key methods you need to practise, from simple one-step equations to equations with brackets and fractions.
线性方程是 KS3 代数的基础。它们以多种不同形式出现，能够自信地解出它们是剑桥 Checkpoint 及更高阶段取得成功的关键。本文涵盖你需要练习的关键方法，从简单的一步方程到带有括号和分数的方程。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is an equation where the unknown, usually written as x, is only raised to the power of 1. This means there are no terms like x², x³ or √x aim of solving a linear equation is to find the value of the unknown that makes the equation true.
线性方程是指未知数（通常用 x 表示）的指数仅为 1 的方程。这意味着没有 x²、x³ 或 √x 这样的项。方程可以简单如 x + 3 = 7，也可以复杂如 3(x − 2) + 5 = 4x − 1。解线性方程的目标就是求出使等式成立的未知数的值。
- Linear equation: 2x + 5 = 11 | 线性方程：2x + 5 = 11
- Not linear: x² + 2 = 6 | 非线性：x² + 2 = 6
- Solution is the value of x | 解就是 x 的值
2. The Balancing Method | 平衡法
The golden rule when solving equations is to keep the balance. Whatever you do to one side of the equation, you must do to the other side. Think of an equation as a set of scales: if you add or subtract a weight on one side, you must add or subtract the same weight on the other side to keep it level. This method allows you to isolate the unknown step by step.
解方程时最重要的法则是保持平衡。你对等式的一边做什么，就必须对另一边也做什么。可以把方程想象成一台天平：如果你在一边加上或减去一个砝码，就必须在另一边加上或减去同样的砝码以保持水平。这个方法能让你一步步把未知数分离出来。
x + 7 = 15
ract 7 from both sides: x + 7 − 7 = 15 − 7, so x = 8. Check: 8 + 7 = 15, which is correct.
两边同时减去 7：x + 7 − 7 = 15 − 7，所以 x = 8。检验：8 + 7 = 15，正确。
3. Solving One-Step and Two-Step Equations | 解一步和两步方程
tract 3 from both sides, then divide both sides by 2. Always perform addition or subtraction before multiplication or division when isolating the variable.
一步方程只需要一次运算就能解出。例如，x − 4 = 9 通过两边同时加 4 来解。两步方程需要两次运算，例如 2x + 3 = 11。首先两边同时减去 3，然后两边同时除以 2。在分离变量时，总是先做加减法，再做乘除法。
2x + 3 = 11
Step 1: 2x + 3 − 3 = 11 − 3, so 2x = 8. Step 2: 2x ÷ 2 = 8 ÷ 2, so x = 4.
第一步：2x + 3 − 3 = 11 − 3，所以 2x = 8。第二步：2x ÷ 2 = 8 ÷ 2，所以 x = 4。
4. Equations with Unknowns on Both Sides | 两边都有未知数的方程
st step is to collect all the variable terms on one side and all the constant terms on the other. Subtract the smaller variable term from both sides to avoid negative coefficients. Then solve the resulting two-step equation as usual.
有些方程两边都有未知数，例如 5x + 2 = 3x + 10。第一步是把所有含未知数的项移到一边，把所有常数项移到另一边。从两边同时减去较小的未知数项，避免出现负系数。然后像往常一样解得到的两步方程。
5x + 2 = 3x + 10
Step 1: 5x − 3x + 2 = 3x − 3x + 10, so 2x + 2 = 10. Step 2: 2x = 8, then x = 4.
第一步：5x − 3x + 2 = 3x − 3x + 10，所以 2x + 2 = 10。第二步：2x = 8，然后 x = 4。
5. Equations with Brackets | 带括号的方程
eful with negative signs, because −(x + 3) means −x − 3, not −x + 3. Once the brackets are removed, collect like terms and solve using the balancing method.
当方程含有括号时，一定要先展开括号。使用分配律：a(b + c) = ab + ac。注意负号，因为 −(x + 3) 表示 −x − 3，而不是 −x + 3。去掉括号后，合并同类项并用平衡法求解。
3(x + 2) = 21
Expand: 3x + 6 = 21. Subtract 6 from both sides: 3x = 15. Divide by 3: x = 5.
展开：3x + 6 = 21。两边同时减去 6：3x = 15。除以 3：x = 5。
6. Equations with Fractions | 带分数的方程
Equations with fractions look intimidating, but they can be simplified quickly by multiplying every term by the lowest common denominator. This clears the fractions and gives a simpler equation with whole numbers. Always multiply both sides by the same value to keep the equation balanced.
带分数的方程看起来有些吓人，但只要把每一项都乘以最小公分母就能快速化简。这样可以消去分数，得到一个更简单的整数方程。一定要两边同时乘以同一个数，保持方程平衡。
x/3 + 4 = 7
Multiply every term by 3: x + 12 = 21. Subtract 12: x = 9.
每一项都乘以 3：x + 12 = 21。减去 12：x = 9。
7. Rearranging Formulas | 公式变形
ake x the subject of y = mx + c, first subtract c, then divide by m, giving x = (y − c)/m.
公式变形使用的规则与解方程相同，但涉及多个变量。例如，要把 a = b + c 中的 b 作为主项，两边同时减去 c 得到 b = a − c。要把 y = mx + c 中的 x 作为主项，先减去 c，再除以 m，得到 x = (y − c)/m。
y = mx + c → x = (y − c)/m
Remember: whatever you do to one side, do to the other. This keeps the formula true for all values.
记住：你对一边做什么，就要对另一边也做什么。这样公式对所有值都成立。
8. Word Problems Leading to Linear Equations | 应用题转化为线性方程
Many KS3 problems ask you to form an equation from a real-life situation. Read the problem carefully, define the unknown with a letter, and translate the words into an equation using operations. Then solve the equation and write the answer in the context of the question, including units where necessary.
许多 KS3 题目要求你根据实际情境列出方程。仔细阅读题目，用字母定义未知数，并用运算把文字转化为方程。然后解方程，并结合题目情境写出答案，必要时加上单位。
Example: A rectangle has length 2x + 1 cm and width 5 cm. Its perimeter is 32 cm. Form an equation and find x.
示例：一个长方形的长为 2x + 1 厘米，宽为 5 厘米。周长为 32 厘米。列出方程并求出 x。
2(2x + 1 + 5) = 32
Simplify: 2(2x + 6) = 32, so 4x + 12 = 32. Subtract 12: 4x = 20. Divide by 4: x = 5.
化简：2(2x + 6) = 32，所以 4x + 12 = 32。减去 12：4x = 20。除以 4：x = 5。
9. Checking Your Solution | 检验你的解
Always substitute your answer back into the original equation to check it works. This is especially important when you have used several steps. If the left-hand side equals the right-hand side after substitution, your solution is correct. If not, go back and look for arithmetic mistakes or sign errors.
一定要把你的答案代回原方程检验是否正确。这在你使用了多个步骤后尤其重要。如果代入后左边等于右边，说明解是正确的。如果不是，就回头检查算术错误或符号错误。
5x + 2 = 3x + 10 with x = 4: LHS = 5(4) + 2 = 22, RHS = 3(4) + 10 = 22. Both sides match.
对于 5x + 2 = 3x + 10，当 x = 4 时：左边 = 5(4) + 2 = 22，右边 = 3(4) + 10 = 22。两边相等。
10. Common Mistakes to Avoid | 常见错误及避免方法
careful when expanding brackets with a negative outside: −2(x − 3) = −2x + 6, not −2x − 6.
一个常见错误是移项时忘记变号。另一个错误是只把左边的一部分除以某个数。一定要对整个边进行除法，而不是只除一个项。另外，展开括号外部为负号时要小心：−2(x − 3) = −2x + 6，而不是 −2x − 6。
- Move terms: change sign | 移项：改变符号
- Divide whole side | 除以整个边
- Negative bracket: distribute sign | 负括号：分配符号
11. Practice Questions Walkthrough | 练习题示范
Let' 3x from both sides: x − 8 = 5, then add 8: x = 13. Check: LHS 4(11) = 44, RHS 39 + 5 = 44.
我们逐步完成三道练习题。首先解 4(x − 2) = 3x + 5。展开得到 4x − 8 = 3x + 5，两边同时减去 3x：x − 8 = 5，然后加 8：x = 13。检验：左边 4(11) = 44，右边 39 + 5 = 44。
− 3 = 5. Multiply every term by 2: x − 6 = 10. Add 6: x = 16. Check: 16/2 − 3 = 8 − 3 = 5.
接下来解 x/2 − 3 = 5。每一项都乘以 2：x − 6 = 10。加 6：x = 16。检验：16/2 − 3 = 8 − 3 = 5。
es: 6 = 3x − 9. Add 9: 15 = 3x. Divide by 3: x = 5. Check: LHS 2(8) = 16, RHS 25 − 9 = 16.
最后解 2(x + 3) = 5x − 9。展开：2x + 6 = 5x − 9。两边同时减去 2x：6 = 3x − 9。加 9：15 = 3x。除以 3：x = 5。检验：左边 2(8) = 16，右边 25 − 9 = 16。
12. Summary and Key Takeaways | 总结与要点
Solving linear equations is all about isolating the unknown using inverse operations while keeping the equation balanced. Work step by step, expand brackets first, clear fractions when needed, and always check your answer by substitution. Practise regularly with different types of equations to build confidence.
解线性方程的核心是利用逆运算分离未知数，同时保持方程平衡。一步一步来，先展开括号，必要时消去分数，并始终通过代入检验答案。通过练习不同类型的方程来建立信心。
| Equation type | Method |
| One-step | Apply one inverse operation |
| Two-step | Add/subtract first, then multiply/divide |
| Unknowns on both sides | Collect like terms on one side |
| With brackets | Expand first |
| With fractions | Multiply by lowest common denominator |
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