📚 Linear Equations and Problem Solving | 线性方程与实际问题求解
In KS3 Cambridge Mathematics, solving linear equations is a central algebraic skill. It allows you to find unknown values in mathematical problems and in real-life situations such as splitting bills, calculating distances or working out unknown measurements.
在 KS3 剑桥数学中，解线性方程是一项核心代数技能。它可以帮助你求出数学问题和实际生活中的未知值，比如分摊账单、计算距离或求出未知的测量数据。
This guide explains how to use the balancing method, solve equations step by step, form equations from word problems, and avoid common errors. Each section includes worked examples and exam tips to build confidence.
本指南将讲解如何使用天平法、逐步解方程、根据文字题列方程以及避免常见错误。每个小节都包括示例和考试技巧，帮助你建立信心。
1. Understanding Linear Equations | 认识线性方程
x + 5 = 13 is linear, while x² + 3 = 7 is not linear because the unknown has a power of 2.
线性方程是指未知数只以 1 次方出现的方程。例如 2x + 5 = 13 是线性方程，而 x² + 3 = 7 不是线性方程，因为未知数的次数为 2。
The expression on the left of the equals sign and the expression on the right are equal in value. This means any operation you perform must keep both sides equal.
等号左边的表达式和右边的表达式在数值上相等。这意味着你所做的任何运算都必须保持两边相等。
In KS3, linear equations can include positive and negative numbers, fractions, decimals, brackets, and unknowns on both sides. Being able to recognise these features helps you choose the best solving strategy.
在 KS3 阶段，线性方程可包含正负数、分数、小数、括号以及两边都有未知数。能够识别这些特征有助于你选择最佳解题策略。
wn that makes the statement true. In x + 3 = 10, the solution is x = 7 because 7 + 3 = 10.
方程的解是使等式成立的未知数的值。在 x + 3 = 10 中，解是 x = 7，因为 7 + 3 = 10。
2. The Balancing Method | 天平法核心思想
Think of an equation as a balance scale. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side to keep the scale balanced.
把方程想象成一架天平。如果你对一边加、减、乘或除以一个数，你必须对另一边做完全相同的运算，才能保持天平平衡。
+ 4 = 9, you can subtract 4 from both sides to isolate x: x + 4 − 4 = 9 − 4, giving x = 5.
例如，如果 x + 4 = 9，你可以两边同时减去 4 来分离 x：x + 4 − 4 = 9 − 4，得到 x = 5。
x + 4 = 9 → x + 4 − 4 = 9 − 4 → x = 5
The balancing method is preferred in Cambridge exams because it shows clear reasoning and reduces the chance of sign mistakes. Never just move a number across without applying the operation to every term.
剑桥考试中更推荐使用天平法，因为它能展示清晰的推理过程，并减少符号出错的可能性。绝不要只是把数字移过去而不对每一项使用该运算。
3. Solving One-Step Equations | 解一步方程
For one-step equations, identify the operation being applied to the unknown and use its inverse operation on both sides. The four basic cases are shown below.
对于一步方程，先识别施加在未知数上的运算，再在两边同时使用它的逆运算。下面列出四种基本情形。
- x + 5 = 12 → subtract 5: x = 7 | x + 5 = 12 → 两边减 5：x = 7
- x − 6 = 4 → add 6: x = 10 | x − 6 = 4 → 两边加 6：x = 10
- 4x = 20 → divide by 4: x = 5 | 4x = 20 → 两边除以 4：x = 5
- x ÷ 3 = 8 → multiply by 3: x = 24 | x ÷ 3 = 8 → 两边乘以 3：x = 24
Always write the operation on both sides rather than simply moving a number across. This helps you avoid losing negative signs or forgetting to apply the operation to every term.
一定要把运算写在两边，而不是简单地把数字移过去。这能帮助你避免丢失负号或忘记对每一项使用该运算。
tive numbers, be careful with signs. Solve −x + 2 = 5 by subtracting 2: −x = 3, so x = −3.
对于负数，要注意符号。解 −x + 2 = 5：先减 2 得 −x = 3
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