Solving Linear Equations | 解线性方程

📚 Solving Linear Equations | 解线性方程

Linear equations are the building blocks of algebra. In KS3 Mathematics, you learn to solve equations where the unknown value is represented by a letter, usually x, and the equation involves only simple operations like addition, subtraction, multiplication, and division. A linear equation can always be written in the form ax + b = c, where a, b, and c are numbers and a ≠ 0. Mastering linear equations gives you the skills to tackle more complex problems in algebra, graphs, and real-world situations. This guide will walk you through every type of linear equation you will encounter at the Cambridge KS3 level, from one‑step to equations with brackets and variables on both sides.

线性方程是代数的基石。在 KS3 数学中，你要学会求解未知数（通常用字母 x 表示）的方程，这些方程只涉及加、减、乘、除等简单运算。线性方程总可以写成 ax + b = c 的形式，其中 a、b、c 是数字且 a ≠ 0。掌握线性方程能让你有能力应对代数、图像和现实情境中更复杂的问题。本指南将带你走过剑桥 KS3 阶段会遇到的每一类线性方程，从一步方程到含有括号和变量在等式两边的方程。

1. Understanding Equations and Expressions | 理解方程与表达式

An equation is a mathematical statement that shows two expressions are equal, connected by an equals sign ‘=’. An expression, on the other hand, is a collection of numbers, variables, and operation symbols without an equals sign. For example, 3x + 5 is an expression, but 3x + 5 = 11 is an equation. When you solve an equation, your goal is to find the value of the variable that makes the equality true. Think of a balance scale: the left side of the equation must have exactly the same value as the right side. Whatever you do to one side, you must do to the other to keep the scales balanced.

方程是一个数学陈述，它表示两个表达式相等，并以等号“=”连接。而表达式则是一组数字、变量和运算符号，没有等号。例如，3x + 5 是一个表达式，但 3x + 5 = 11 是一个方程。解方程时，你的目标是找到使等式成立的变量的值。可以想象成一架天平：方程左边的值必须精确等于右边的值。无论你对一边做什么，都必须同时对另一边做同样的操作，以保持天平平衡。

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2. Solving One-Step Equations (Addition & Subtraction) | 解一步方程（加减）

The simplest equations only require one operation to isolate the variable. If the variable has something added to it, subtract that amount from both sides. If something is subtracted from the variable, add it to both sides. For example, to solve x + 7 = 15, subtract 7 from both sides: x + 7 − 7 = 15 − 7, which simplifies to x = 8. Always present your solution as x = … and ensure you keep the variable positive. Remember the golden rule: do exactly the same operation on both sides of the equals sign.

最简单的方程只需一步运算就能分离出变量。如果变量被加上一个数，两边同时减去这个数。如果变量被减去一个数，两边同时加上这个数。例如，解 x + 7 = 15，两边同时减去 7：x + 7 − 7 = 15 − 7，化简得 x = 8。始终将解表示为 x = …，并确保变量为正。牢记黄金法则：对等号两边做完全相同的运算。

x − 9 = 3 → x − 9 + 9 = 3 + 9 → x = 12

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3. Solving One-Step Equations (Multiplication & Division) | 解一步方程（乘除）

When the variable is multiplied by a number, you divide both sides by that number to find the unknown. For a division equation like x/4 = 5, multiply both sides by 4 to clear the fraction. Consider 3x = 18. Since 3 is multiplied by x, divide both sides by 3: 3x/3 = 18/3, giving x = 6. If the equation is x/6 = 2, multiply both sides by 6: (x/6) × 6 = 2 × 6, so x = 12. Always write the division or multiplication explicitly and simplify where possible.

当变量乘以一个数时，两边同时除以这个数就能求出未知数。对于像 x/4 = 5 这样的除法方程，两边同时乘以 4 来消去分数。考虑 3x = 18。因为 3 与 x 相乘，两边同时除以 3：3x/3 = 18/3，得到 x = 6。如果方程是 x/6 = 2，两边同时乘以 6：(x/6) × 6 = 2 × 6，所以 x = 12。始终明确写出除法或乘法，并尽可能化简。

4x = 20 → x = 5

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4. Introducing Two-Step Equations | 两步方程入门

A two-step equation involves two operations to isolate the variable. A typical form is ax + b = c. You must first undo the addition or subtraction, then undo the multiplication or division. For instance, solve 2x + 3 = 11. First subtract 3 from both sides: 2x + 3 − 3 = 11 − 3, which gives 2x = 8. Next, divide both sides by 2: 2x/2 = 8/2, so x = 4. Always perform the addition/subtraction step before the multiplication/division step, reversing the order of operations (BIDMAS in reverse).

两步方程需要两步运算来分离变量。典型形式为 ax + b = c。你必须先消去加法或减法，再消去乘法或除法。例如，解 2x + 3 = 11。首先两边同时减去 3：2x + 3 − 3 = 11 − 3，得到 2x = 8。接着两边同时除以 2：2x/2 = 8/2，所以 x = 4。始终先进行加减步骤，再进行乘除步骤，反向遵循运算顺序（逆 BIDMAS）。

5x − 2 = 13 → 5x = 15 → x = 3

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5. Solving Equations with Negative Coefficients | 解带负系数的方程

When the variable term has a negative coefficient, such as −4x, you can either divide by the negative number or first add/subtract to make the term positive. For −3x + 2 = 14, subtract 2 from both sides: −3x = 12. Then divide both sides by −3: x = −4. Always be careful with signs: a negative divided by a negative gives a positive. If the equation is 7 − 2x = 1, rearrange to isolate the x term: subtract 7 from both sides gives −2x = −6, then divide by −2 to get x = 3. Alternatively, you can add 2x to both sides to avoid starting with a negative.

当变量项有负系数时，例如 −4x，你可以除以这个负数，也可以先通过加减让该项变正。对于 −3x + 2 = 14，两边减去 2：−3x = 12。然后两边除以 −3：x = −4。始终注意符号：负数除以负数得正数。如果方程是 7 − 2x = 1，重新整理以分离 x 项：两边减去 7 得 −2x = −6，然后除以 −2 得 x = 3。另一种方法是两边同时加上 2x，避免一开始就面对负数。

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6. Equations with Brackets | 带括号的方程

If an equation contains brackets, you must expand (multiply out) the brackets first. Use the distributive law: a(b + c) = ab + ac. For example, solve 3(x + 4) = 21. Expand to get 3x + 12 = 21. Then follow the two-step method: subtract 12 from both sides to get 3x = 9, and divide by 3 to find x = 3. In some cases you may have a negative sign before the bracket, e.g. 2 − 3(x − 1) = 5. Distribute the −3: −3 × x = −3x and −3 × −1 = +3, so the equation becomes 2 − 3x + 3 = 5, then simplify to 5 − 3x = 5. Always expand carefully, especially with negative multipliers.

如果方程包含括号，你必须先展开（乘出）括号。使用分配律：a(b + c) = ab + ac。例如，解 3(x + 4) = 21。展开得 3x + 12 = 21。然后按照两步法：两边减去 12 得到 3x = 9，再除以 3 得 x = 3。有些情况下，括号前可能有负号，例如 2 − 3(x − 1) = 5。分配 −3：−3 × x = −3x，−3 × −1 = +3，于是方程变为 2 − 3x + 3 = 5，化简得 5 − 3x = 5。始终仔细展开，尤其是有负乘数时。

2(4x − 3) = 10 → 8x − 6 = 10 → 8x = 16 → x = 2

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7. Equations with Variables on Both Sides | 变量在等式两边的方程

Sometimes the unknown x appears on both the left-hand side (LHS) and the right-hand side (RHS). The strategy is to collect all variable terms on one side and the constant numbers on the other. For example, 5x + 2 = 3x + 8. Subtract 3x from both sides: 5x − 3x + 2 = 3x − 3x + 8 → 2x + 2 = 8. Then subtract 2 from both sides: 2x = 6, so x = 3. If the equation is 4x − 7 = x + 5, subtract x from both sides to get 3x − 7 = 5, add 7 to both sides: 3x = 12, x = 4. Always aim to have a positive coefficient for x on one side. You can choose whichever side gives a simpler expression.

有时未知数 x 同时出现在左边（LHS）和右边（RHS）。策略是将所有变量项集中到一边，常数项集中到另一边。例如，5x + 2 = 3x + 8。两边减去 3x：5x − 3x + 2 = 3x − 3x + 8 → 2x + 2 = 8。然后两边减去 2：2x = 6，所以 x = 3。如果方程是 4x − 7 = x + 5，两边减去 x 得到 3x − 7 = 5，两边加上 7：3x = 12，x = 4。始终致力于让 x 在一边有正系数。你可以选择让表达式更简单的一边。

2x + 9 = 6x − 3 → 9 + 3 = 6x − 2x → 12 = 4x → x = 3

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8. Checking Your Solution | 检验解

You can always verify your answer by substituting the value back into the original equation. This is an essential habit that identifies mistakes. For x = 3 in the equation 2x + 9 = 6x − 3, the left side becomes 2(3) + 9 = 6 + 9 = 15, and the right side becomes 6(3) − 3 = 18 − 3 = 15. Both sides equal 15, so the solution is correct. Checking also reinforces your understanding of how equations work. At KS3, you may be asked to show your check in your working.

你总可以将解代入原方程来检验答案。这是发现错误的重要习惯。对于方程 2x + 9 = 6x − 3 的解 x = 3，左边变为 2(3) + 9 = 6 + 9 = 15，右边变为 6(3) − 3 = 18 − 3 = 15。两边都等于 15，所以解是正确的。检验还能加深你对方程运作方式的理解。在 KS3 阶段，你可能需要在解答过程中展示检验步骤。

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9. Common Mistakes to Avoid | 常见错误

Many students make avoidable errors when solving linear equations. One common mistake is forgetting to multiply every term inside the brackets when expanding, especially with a negative sign outside. Another is performing operations in the wrong order, such as dividing before adding or subtracting. Also, when moving terms across the equals sign, some students change the sign incorrectly. Remember, subtracting a term is the same as adding its opposite. Avoid writing the solution as a mixture like ‘x = 4 = answer’ — just write ‘x = 4’. Lastly, always check that your final answer is a logical number and satisfies the original equation.

很多学生在解线性方程时会犯可以避免的错误。一个常见错误是展开括号时忘记乘括号内的每一项，尤其在括号外有负号时。另一个错误是运算顺序不对，比如在加减之前先做了除法。还有，在将项移到等号另一边时，有些学生错误地改变符号。记住，减去一项等价于加上它的相反数。不要将解写成 “x = 4 = 答案” 这样的混合形式——只写 “x = 4” 即可。最后，始终检查最终答案是否为合理的数字，并满足原方程。

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10. Real-life Applications | 实际应用

Linear equations are not just abstract exercises; they model many real-world situations. For example, if a mobile phone plan costs £10 per month plus £0.05 per text, and you have a budget of £25, the equation 0.05t + 10 = 25 helps you find how many texts t you can send. Solving gives t = 300 texts. Tickets for a school play: adult tickets cost £8, children £4, and total sales are £200 from 40 tickets. Setting up the equation 8a + 4(40 − a) = 200 allows you to find the number of adult tickets a. Being able to translate a word problem into an equation is a key skill in KS3 mathematics.

线性方程不仅是抽象的练习，它们可以模拟许多现实情境。例如，某手机套餐每月收费 10 英镑，每条短信 0.05 英镑，你的预算是 25 英镑，方程 0.05t + 10 = 25 可以帮你算出能发送的短信数量 t。解方程得 t = 300 条短信。学校话剧门票：成人票 8 英镑，儿童票 4 英镑，40 张票的总销售额为 200 英镑。列出方程 8a + 4(40 − a) = 200，就能求出成人票的数量 a。能将文字题转化为方程是 KS3 数学的一项关键技能。

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11. Practice Tips for Mastery | 精通练习技巧

To become confident in solving linear equations, practice regularly and challenge yourself with a variety of types. Start with simple one-step equations, then move to two-step, brackets, and variables on both sides. Write out each step clearly, showing the operation on both sides. Use a checking routine to confirm your solution. When you encounter a tough equation, write it on a separate piece of paper and try to solve it step by step without looking at the solution. Mixed practice, where you don’t know the type of equation in advance, is particularly useful for exams. You can also create your own equations and swap with a friend to solve.

要自信地解线性方程，需要定期练习并挑战各种类型的题目。从简单的一步方程开始，然后过渡到两步、带括号和变量在等式两边的方程。清晰地写出每一步，显示两边的运算。采用检验程序来确认解。遇到棘手的方程时，把它写在另一张纸上，不看书上的答案，尝试逐步求解。混合练习（即事先不知道方程的类型）对考试特别有用。你还可以自己编方程，和朋友交换解答。

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12. Summary and Key Points | 总结与要点

Solving linear equations is about isolating the unknown variable while maintaining balance. Remember the reverse BIDMAS order: undo addition/subtraction first, then multiplication/division. Always apply the same operation to both sides. When brackets are present, expand first. If the variable appears on both sides, collect like terms on one side. Check your answer by substitution. Master these techniques, and you will build a strong algebraic foundation for your future studies in mathematics. Keep a clear layout in your working, and never rush.

解线性方程的核心是在保持平衡的同时分离出未知变量。记住逆 BIDMAS 顺序：先消去加减，再消去乘除。始终对两边做相同的运算。有括号时，先展开。如果变量出现在两边，将同类项集中到一边。通过代入检验答案。掌握这些技巧，你将为未来的数学学习打下坚实的代数基础。解题过程中保持清晰的布局，切勿急躁。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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