Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Welcome to this revision guide on solving linear equations, a key topic in the Cambridge Lower Secondary Mathematics curriculum. Often featured on worksheets found on page 28 of your practice book, linear equations form the backbone of algebra and appear in countless real-life situations. Understanding how to isolate the unknown variable and find its value is a skill that will support your learning through KS3, IGCSE, and beyond. In this article, we will break down the process step by step, from simple one-step equations to those involving brackets and fractions, and show you how to avoid common mistakes.

欢迎阅读本篇关于解一元一次方程的复习指南，这是剑桥初中数学课程的核心主题之一。一元一次方程常常出现在练习册第28页等习题中，它们是代数的基石，也广泛存在于现实生活场景中。掌握如何分离未知数并求出它的值，将支撑你在KS3、IGCSE乃至更高阶段的学习。在本文中，我们将逐步拆解过程，从简单的一步方程到含有括号和分数的方程，并告诉你如何避免常见错误。

1. What is a Linear Equation? | 什么是线性方程？

A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b, and c are numbers and x is the unknown. The word ‘linear’ comes from the fact that its graph is a straight line. In KS3, we focus on finding the value of the variable that makes the equation true. For example, 2x + 3 = 11 is a linear equation. The solution is the number that, when substituted for x, gives a true statement.

一元一次方程是指可以写成 ax + b = c 形式的方程，其中 a、b 和 c 是数字，x 是未知数。“线性”一词源于它的图像是一条直线。在KS3阶段，我们的重点是求出使方程成立的未知数的值。例如, 2x + 3 = 11 就是一个线性方程。解就是代入 x 后能使等式成立的数。

2. The Balance Method | 天平法

Imagine an old-fashioned balance scale. Whatever you do to one side, you must do exactly the same to the other side to keep it level. This is the golden rule of solving equations. If you add, subtract, multiply, or divide one side of an equation, you must perform the same operation on the other side. Using the balance method helps you visualise the process and avoid mistakes.

想象一个老式天平。无论你对一边做了什么，都必须对另一边做完全相同的操作才能保持平衡。这就是解方程的金科玉律。如果你在方程的一边加、减、乘或除，你必须对另一边进行同样的运算。使用天平法可以帮助你形象化地理解过程，避免出错。

3. Solving by Inverse Operations | 用逆运算求解

The simplest equations require only one inverse operation. To solve x + 5 = 12, subtract 5 from both sides: x = 7. For x − 3 = 8, add 3 to both sides. When the variable is multiplied, like 4x = 20, divide both sides by 4. When it is divided, like x / 6 = 3, multiply both sides by 6. Always write the result as x = …

最简单的方程只需一步逆运算即可求解。对于 x + 5 = 12，两边同时减去5，得到 x = 7。对于 x − 3 = 8，两边同时加3。当未知数被乘时，如 4x = 20，两边同时除以4。当未知数被除时，如 x / 6 = 3，两边同时乘6。最后结果总是写成 x = … 的形式。

Example: 5x = 35 → x = 35 ÷ 5 → x = 7

4. Solving Two-Step Equations | 解两步方程

When an equation involves more than one operation, undo them in reverse order of BIDMAS/BODMAS. For example, to solve 2x + 3 = 11, first subtract 3 from both sides, then divide by 2. Writing each step clearly above an arrow or on a new line helps you stay organised.

当方程包含不止一种运算时，要按照BIDMAS/BODMAS的逆顺序逐步消去。例如，解 2x + 3 = 11，先从两边减去3，再除以2。把每一步清楚地写在箭头上方或另起一行，有助于保持思路清晰。

2x + 3 = 11
2x = 11 − 3 → 2x = 8
x = 8 ÷ 2 → x = 4

5. Equations with the Unknown on Both Sides | 未知数在等式两边的方程

Sometimes the variable appears on both sides, e.g. 5x + 2 = 3x + 10. The strategy is to collect the variable terms on one side and the numbers on the other. Subtract the smaller variable term from both sides: 5x − 3x + 2 = 3x − 3x + 10 gives 2x + 2 = 10. Then solve the resulting two-step equation.

有时未知数同时出现在等式两边，例如 5x + 2 = 3x + 10。策略是将含未知数的项移到一边，数字移到另一边。从两边减去较小的未知数项：5x − 3x + 2 = 3x − 3x + 10 得到 2x + 2 = 10。然后解所得的两步方程。

Alternatively, you can move all terms to one side by adding or subtracting strategically. Always remember to keep the balance.

或者，你可以策略性地加减，把所有项移到同一边。始终牢记保持平衡。

6. Equations with Brackets | 带括号的方程

When brackets are present, expand them first using the distributive law. For 3(x + 4) = 21, multiply both terms inside the bracket by 3 to get 3x + 12 = 21. Then subtract 12 and divide by 3. If the bracket has a negative coefficient, be careful with signs.

当方程中含有括号时，先用分配律展开。对于 3(x + 4) = 21，将括号内每一项都乘3，得到 3x + 12 = 21。然后减去12，再除以3。如果括号前的系数是负数，要特别注意符号。

−2(x − 5) = 6 → −2x + 10 = 6 → −2x = −4 → x = 2

7. Equations with Fractions | 含分数的方程

Fractions can make equations look tricky, but multiplying every term by the common denominator eliminates them. For an equation like x/3 + 2 = 5, multiply all terms by 3 to obtain x + 6 = 15. For (x + 1)/2 = (x − 3)/4, multiply both sides by 4, or cross-multiply, giving 2(x + 1) = x − 3.

分数会让方程看起来棘手，但将每一项都乘上公分母就可以消去分母。对于 x/3 + 2 = 5，将所有项都乘3，得到 x + 6 = 15。对于 (x + 1)/2 = (x − 3)/4，两边同乘4，或交叉相乘，得到 2(x + 1) = x − 3。

Always apply the multiplication to every term, including whole numbers. 务必将乘法应用到每一项，包括整数项。
Simplify fractions where possible before starting to reduce numbers. 在开始前尽可能先约分，以减小数字。

8. Forming Equations from Word Problems | 从文字题建立方程

Real-life problems require you to translate English (or your language) into algebra. Look for keywords: ‘sum of’ means addition, ‘product of’ means multiplication, ‘is’ or ‘equals’ means the = sign. For example, “Three more than twice a number is 15” becomes 2x + 3 = 15. Define the variable first, then write the equation, and finally solve and check.

实际问题需要你将文字语言转化为代数。寻找关键词：‘和’表示加法，‘积’表示乘法，‘是’或‘等于’表示等号。例如，“一个数的两倍再加3等于15”转化为 2x + 3 = 15。先设未知数，再列出方程，最后求解并检验。

Practice forming equations from descriptions and solving them. This skill is frequently tested in Cambridge KS3 assessments.

多加练习根据描述列方程并求解。这项技能在剑桥KS3测验中经常考查。

9. Checking Your Solution | 检验解

Once you find a value for x, always substitute it back into the original equation to verify. For 4x − 7 = 13, you get x = 5. Check: left side = 4(5) − 7 = 20 − 7 = 13, which equals the right side. If your check fails, go back and find the error. This habit can save you many marks in exams.

求出 x 的值后，一定要代回原方程进行验证。对于 4x − 7 = 13，得到 x = 5。检验：左边 = 4(5) − 7 = 20 − 7 = 13，等于右边。如果检验不通过，就回去找出错误。养成这个习惯可以在考试中为你保住很多分数。

10. Common Mistakes to Avoid | 需要避免的常见错误

Even confident students slip up. Watch out for these typical errors:

即使有信心的学生也会出错。留意以下常见错误：

Forgetting to do the same operation on both sides. 忘记在两边执行相同操作。
Incorrect sign when moving a term (e.g., 5 − x = 2 becomes −x = −3, not x = 3 without the sign change). 移项时符号错误（如 5 − x = 2 应为 −x = −3，而不是没变号直接得 x = 3）。
Expanding brackets incorrectly, especially when a negative sign is outside. 括号展开错误，特别是括号外有负号时。
Multiplying only the fractional term by the denominator instead of the whole equation. 只将分母乘上分数项，而非整个方程。
Not simplifying before solving, making numbers larger than necessary. 解题前没有先化简，导致数字不必要地变大。

Read each equation carefully and take your time. Accuracy is more important than speed.

仔细阅读每一个方程，慢慢来。准确比速度更重要。

11. Algebraic Manipulation Warm-up | 代数变形热身

Before tackling mixed equations, it’s useful to practise algebraic simplification. Collect like terms: 5y + 3 − 2y + 7 becomes 3y + 10. Simplify expressions like 2(3a − 4) + 5a to 6a − 8 + 5a = 11a − 8. These skills make solving equations much smoother.

在处理混合方程之前，练习代数化简很有帮助。合并同类项：5y + 3 − 2y + 7 化简为 3y + 10。化简像 2(3a − 4) + 5a 的式子，得到 11a − 8。这些技巧能让解方程更顺畅。

12. Practice Problems and Final Tips | 练习题与最后建议

Try these as a quick self-check:

试着用这些题快速自测：

Solve 3x + 8 = 20 解方程 3x + 8 = 20
Solve 7 − 2x = 1 解方程 7 − 2x = 1
Solve 4(x − 3) = 2x + 6 解方程 4(x − 3) = 2x + 6
Solve (x + 2)/3 = (x − 1)/2 解方程 (x + 2)/3 = (x − 1)/2

If you can solve these confidently, you’re well prepared for page 28 exercises and beyond. Remember that practice is the key. Work through problems step by step, and don’t hesitate to write down every single line of working. With consistent practice, linear equations will become second nature.

如果你能自信地解出这些题，你就已经为第28页及以后的练习做好了充分准备。记住，练习是关键。一步一步地解题，不要犹豫，要把每一步都写下来。持续练习，一元一次方程将成为你的本能。

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Solving Linear Equations | 解一元一次方程