📚 KS3 Cambridge Mathematics: Solving Linear Equations | KS3 剑桥数学：解线性方程
Linear equations are one of the most important building blocks in KS3 mathematics. They appear in many topics, from algebra and problem-solving to graphs and real-life situations. In this article, we will explore how to solve linear equations step by step, using clear methods and examples that match the Cambridge KS3 curriculum.
线性方程是 KS3 数学中最重要的基础模块之一。它们出现在许多主题中，从代数、解题到图像和现实情境。本文将逐步探讨如何解线性方程，使用符合剑桥 KS3 课程要求的清晰方法和示例。
1. What is a Linear Equation? | 什么是线性方程？
n because the highest power of x is 1. The graph of a linear equation is always a straight line, which is why it is called 'linear'.
线性方程是指未知变量（通常写作 x）的指数为 1 的方程。这意味着方程中没有 x² 或 x³ 这样的项。例如，2x + 3 = 11 是一个线性方程，因为 x 的最高次幂是 1。线性方程的图像总是一条直线，这就是它被称为 '线性' 的原因。
2x + 3 = 11
x + 5 = 12
2. Key Vocabulary | 关键术语
Before solving equations, it helps to know the key terms. The left-hand side (LHS) is the expression to the left of the equals sign, and the right-hand side (RHS) is the expression to the right. A solution is the value of x that makes the equation true. Inverse operations are opposite operations, such as addition and subtraction, or multiplication and division. A constant is a number on its own, while a coefficient is the number multiplying the variable.
在解方程之前，了解关键术语会很有帮助。左边（LHS）是等号左边的表达式，右边（RHS）是等号右边的表达式。解是使方程成立的 x 值。逆运算是相反的运算，例如加法和减法，或乘法和除法。常数是单独出现的数字，而系数是乘以变量的数字。
- In 3x + 4 = 10, the LHS is 3x + 4 and the RHS is 10.
- In 3x + 4 = 10, 3 is the coefficient of x, and 4 is the constant.
- In 3x + 4 = 10, the solution is x = 2 because 3 × 2 + 4 = 10.
在 3x + 4 = 10 中，左边是 3x + 4，右边是 10。
在 3x + 4 = 10 中，3 是 x 的系数，4 是常数。
在 3x + 4 = 10 中，解是 x = 2，因为 3 × 2 + 4 = 10。
3. The Balance Method | 天平法
The balance method is a visual way to understand equations. Imagine an old-fashioned balance scale. The equals sign means both sides are perfectly balanced. Whatever operation you do to one side, you must do the same to the other side to keep the balance. This is the golden rule of solving equations.
天平法是一种理解方程的直观方法。想象一台老式天平。等号意味着两边完全平衡。无论你对一边进行什么运算，都必须对另一边进行同样的运算，以保持平衡。这是解方程的黄金法则。
If you add, subtract, multiply, or divide one side by a number, you must do exactly the same to the other side. This rule keeps the equation true and helps you isolate the unknown variable.
如果你对一边进行加、减、乘或除以某个数，你必须对另一边进行完全相同的操作。这条规则使方程保持成立，并帮助你分离出未知变量。
4. Solving One-Step Equations | 解一步方程
4x = 20, divide both sides by 4: x = 5. For x − 3 = 9, add 3 to both sides to get x = 12.
一步方程只需要一个逆运算即可求解。例如，解 x + 7 = 15，两边减去 7：x + 7 − 7 = 15 − 7，因此 x = 8。类似地，解 4x = 20，两边除以 4：x = 5。对于 x − 3 = 9，两边加上 3 得到 x = 12。
x + 7 = 15 → x = 15 − 7 → x = 8
4x = 20 → x = 20 ÷ 4 → x = 5
, such as −x = 7, multiply both sides by −1 to find x = −7. Remember that −x means −1 × x.
如果系数为负，例如 −x = 7，两边乘以 −1 得到 x = −7。请记住 −x 表示 −1 × x。
5. Solving Two-Step Equations | 解两步方程
sides by 3: x = 5. Always undo addition or subtraction before multiplication or division.
两步方程涉及两个运算。例如，解 3x − 4 = 11。首先，两边加上 4：3x − 4 + 4 = 11 + 4，得到 3x = 15。然后两边除以 3：x = 5。一定要先处理加减，再处理乘除。
iply both sides by 2: x = 14. Work backwards through the order of operations to isolate x.
另一个例子是 x ÷ 2 + 6 = 13。首先两边减去 6：x ÷ 2 = 7。然后两边乘以 2：x = 14。按照运算顺序倒推，以分离出 x。
6. Equations with Brackets | 含有括号的方程
y divide by 2: x = 4. You can check by substituting x = 4 back into the original equation.
如果方程含有括号，应先把括号展开。例如，2(x + 3) = 14。展开括号：2x + 6 = 14。然后两边减去 6：2x = 8。最后除以 2：x = 4。你可以把 x = 4 代回原方程进行检验。
th sides: 3x = 27. Divide by 3: x = 9. Be careful with negative signs inside the brackets.
对于 3(x − 5) = 12，展开得到 3x − 15 = 12。两边加上 15：3x = 27。除以 3：x = 9。注意括号内的负号。
7. Equations with Unknowns on Both Sides | 未知数在两侧的方程
h sides: 2x + 2 = 10. Then subtract 2 from both sides: 2x = 8. Finally divide by 2: x = 4.
当 x 出现在两边时，首先把含 x 的项移到同一边。例如，5x + 2 = 3x + 10。两边减去 3x：2x + 2 = 10。然后两边减去 2：2x = 8。最后除以 2：x = 4。
= 15. Divide by 5: x = 3. Choose the side with the larger coefficient to keep x positive.
如果方程是 7x − 3 = 2x + 12，两边减去 2x：5x − 3 = 12。两边加上 3：5x = 15。除以 5：x = 3。选择系数较大的一边来保持 x 为正。
8. Checking Your Solution | 检验解
ecomes 2(4 + 3) = 2 × 7 = 14, which equals the RHS. This confirms the solution is correct.
一定要把答案代回原方程进行检验。对于 2(x + 3) = 14 中的 x = 4，左边变为 2(4 + 3) = 2 × 7 = 14，与右边相等。这证明解是正确的。
Checking helps you catch small arithmetic errors. Write out the substitution clearly, showing each step, so you can identify exactly where any mistake occurred.
检验有助于发现小的算术错误。清晰地写出代入过程，展示每一步，这样你就能准确找到错误发生的位置。
9. Common Mistakes | 常见错误
g only one term by a number instead of the whole side. Always treat each side as a complete expression.
一个常见错误是忘记对两边进行相同的运算。另一个错误是负数处理不当，例如 x − 5 = 3 意味着 x = 8，而不是 x = −2。第三个错误是只把一个项除以某个数，而没有对整边进行除法。应始终把每一边看作一个完整的表达式。
| Mistake | Wrong | Correct |
| Forgetting to do the same to both sides | x + 3 = 7 → x = 7 | x + 3 = 7 → x = 4 |
| Incorrect negative signs | x − 5 = 3 → x = −2 | x − 5 = 3 → x = 8 |
| Dividing only part of a side | 2x + 6 = 10 → x + 6 = 5 | 2x + 6 = 10 → 2x = 4 → x = 2 |
错误：忘记对两边进行相同操作。错误：负数符号不正确。错误：只除以一边的一部分。正确：完整地除以每一项。
10. Practice Questions | 练习题
Try these questions to test yourself. Solve each equation and check your answer by substitution.
尝试以下题目来检验自己。解每个方程并通过代入检验你的答案。
- Solve x + 9 = 17
- Solve 7x = 42
- Solve 4x − 5 = 11
- Solve 3(x − 2) = 18
- Solve 6x + 1 = 2x + 17
- Solve 5 − 2x = x + 11
= −2. Work through the steps slowly and use the balance method to explain your reasoning.
答案：x = 8，x = 6，x = 4，x = 8，x = 4，x = −2。慢慢完成每个步骤，并使用天平法解释你的推理。
11. Real-Life Applications | 实际应用
g costs, distances, or unknown measurements. For example, if a taxi charges a fixed fee of £3 plus £2 per mile and the total fare is £15, the equation 2m + 3 = 15 can be solved to find m = 6 miles.
线性方程可用于解决实际问题，如计算费用、距离或未知测量值。例如，如果出租车收取固定费用 3 英镑，每英里 2 英镑，总费用为 15 英镑，则方程 2m + 3 = 15 可解出 m = 6 英里。
her example is working out how many weeks you need to save pocket money. If you start with £10 and save £5 per week, and you need £45, the equation 5w + 10 = 45 gives w = 7 weeks.
另一个例子是计算你需要存多少周的零用钱。如果你从 10 英镑开始，每周存 5 英镑，你需要 45 英镑，方程 5w + 10 = 45 得出 w = 7 周。
12. Summary | 总结
To solve linear equations confidently, remember to use inverse operations, keep the balance, expand brackets first, collect like terms, and always check your answer. Regular practice will make these steps automatic and prepare you for more advanced algebra.
要自信地解线性方程，请记住使用逆运算、保持平衡、先展开括号、合并同类项，并始终检验答案。定期练习将使这些步骤变得自动化，并为更高级的代数做好准备。
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