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 Cambridge Mathematics, solving equations step by step helps you develop logical thinking and problem-solving skills. Whether you are tackling one-step problems or equations with unknowns on both sides, mastering these methods will prepare you for more advanced topics.

线性方程是代数的基石。在KS3剑桥数学课程中，逐步解方程能帮助你培养逻辑思维和解题能力。无论你面对的是简单的一步方程还是两边都含有未知数的复杂方程，掌握这些方法都将为你学习更深入的内容做好准备。

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1. What is an Equation? | 什么是方程？

An equation is a mathematical statement that shows two expressions are equal, using the equals sign ‘=’. For example, 3x + 2 = 11. The left-hand side (LHS) is 3x + 2, and the right-hand side (RHS) is 11. Both sides represent the same value when x is the solution.

方程是表明两个表达式相等的数学陈述，使用等号“=”连接。例如，3x + 2 = 11。左边（LHS）是3x + 2，右边（RHS）是11。当x取解的值时，两边表示相同的数值。

The goal of solving an equation is to find the value of the unknown variable, usually written as a letter like x, y or a. In an equation, the variable can appear once or several times, but the equality must always hold true.

解方程的目标是找出未知变量的值，变量通常用x、y或a这样的字母表示。在方程中，变量可能出现一次或多次，但等号两边的相等关系必须始终成立。

Example: x + 3 = 7

例子：x + 3 = 7

Here x = 4 because 4 + 3 = 7. We say that 4 is the solution or the root of the equation.

这里x = 4，因为4 + 3 = 7。我们说4是这个方程的解或根。

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2. Balance Method | 天平法

Think of an equation as a balanced scale. Whatever you do to one side, you must do exactly the same to the other side to keep the scale balanced. This idea is called the balance method.

把方程想象成一个平衡的天平。无论你对其中一边做什么操作，都必须在另一边做完全相同的操作，以保持天平平衡。这个思路就是天平法。

Operations include adding, subtracting, multiplying or dividing by the same number or expression on both sides. The balance method ensures that the equality remains true throughout the solving process.

这些操作包括在两边同时加上、减去、乘以或除以同一个数或表达式。天平法确保在求解过程中等号两边的相等关系始终成立。

For example, to solve x + 5 = 12, we subtract 5 from both sides: x + 5 − 5 = 12 − 5, which simplifies to x = 7.

例如，要解 x + 5 = 12，我们在两边同时减去5：x + 5 − 5 = 12 − 5，简化后得到 x = 7。

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3. One-Step Equations | 一步方程

A one-step equation requires just one operation to isolate the variable. The four basic forms involve addition, subtraction, multiplication and division.

一步方程只需进行一次运算就能把变量分离出来。四种基本形式分别涉及加法、减法、乘法和除法。

Addition form: x + 4 = 9 → subtract 4: x = 5.
Subtraction form: y − 3 = 6 → add 3: y = 9.
Multiplication form: 5a = 20 → divide by 5: a = 4.
Division form: b ÷ 2 = 7 → multiply by 2: b = 14.

加法形式：x + 4 = 9 → 减去4：x = 5。
减法形式：y − 3 = 6 → 加上3：y = 9。
乘法形式：5a = 20 → 除以5：a = 4。
除法形式：b ÷ 2 = 7 → 乘以2：b = 14。

Always write the operation you perform on both sides clearly, so you can trace your steps if you make a mistake. One-step equations are the quickest to solve but form the basis for more complicated problems.

一定要清楚地写出你在两边进行的运算，这样即使出错也能追溯你的步骤。一步方程解起来最快，但它们构成了更复杂问题的基础。

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4. Two-Step Equations | 两步方程

Two-step equations involve two different operations. You must undo them in the reverse order of BIDMAS (or PEMDAS). First, deal with addition or subtraction, then handle multiplication or division.

两步方程涉及两种不同的运算。你必须按照与运算顺序（如BIDMAS）相反的顺序逆向操作。首先处理加减，然后处理乘除。

Solve 2x + 5 = 13

解方程 2x + 5 = 13

Step 1: Subtract 5 from both sides → 2x = 8
Step 2: Divide both sides by 2 → x = 4

步骤1：两边同时减去5 → 2x = 8
步骤2：两边同时除以2 → x = 4

If the term with the variable is subtracted, such as 3y − 7 = 8, add 7 first, then divide by 3. Always keep the balance method in mind – the same operation on both sides.

如果含变量的项是被减去的，比如 3y − 7 = 8，先加上7，再除以3。始终牢记天平法——在两边进行相同的运算。

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5. Equations with Brackets | 带括号的方程

When an equation contains brackets, the first step is usually to expand them. Multiply each term inside the bracket by the number or term outside.

当方程含有括号时，第一步通常是展开括号。将括号内的每一项都乘以外面的数或式子。

Example: 2(3x + 4) = 20

例子：2(3x + 4) = 20

Expand: 2 × 3x + 2 × 4 = 6x + 8. So the equation becomes 6x + 8 = 20.
Then solve: 6x = 12 → x = 2.

展开：2 × 3x + 2 × 4 = 6x + 8。于是方程变为 6x + 8 = 20。
接着求解：6x = 12 → x = 2。

Sometimes the bracket has a minus sign in front, like −(2x − 3) = 5. This is equivalent to −1 multiplied by (2x − 3), giving −2x + 3 = 5. Always take care with negative signs.

有时括号前面是负号，比如 −(2x − 3) = 5。这相当于用-1去乘(2x − 3)，得到 −2x + 3 = 5。处理负号时一定要格外小心。

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6. Equations with Unknowns on Both Sides | 两边都有未知数的方程

If the variable appears on both sides of the equals sign, collect all variable terms on one side and all constant terms on the other. This is often done by adding or subtracting terms from both sides.

如果变量出现在等号的两边，就要把所有含变量的项集中到一边，所有常数项集中到另一边。通常通过在两边加或减项来实现。

Solve 5x + 2 = 3x + 10

解方程 5x + 2 = 3x + 10

Step 1: Subtract 3x from both sides → 2x + 2 = 10
Step 2: Subtract 2 from both sides → 2x = 8
Step 3: Divide by 2 → x = 4

步骤1：两边同时减去3x → 2x + 2 = 10
步骤2：两边同时减去2 → 2x = 8
步骤3：除以2 → x = 4

You can also bring the variable terms to the right-hand side. The choice doesn’t change the solution, but it’s a good habit to aim for a positive coefficient of the variable.

你也可以把变量项移到右边。选择哪一边不会改变解，但养成让变量系数为正的习惯会让过程更清晰。

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7. Fractional Equations | 含分数的方程

Equations with fractions can be solved by clearing the fractions. Multiply every term on both sides by the least common multiple (LCM) of the denominators. Then solve the resulting equation without fractions.

含有分数的方程可以通过去分母来求解。用分母的最小公倍数（LCM）去乘两边的每一项，然后解这个没有分数的方程。

Solve x/2 + 3 = 5

解方程 x/2 + 3 = 5

Multiply every term by 2: x + 6 = 10 → x = 4.

每一项都乘以2：x + 6 = 10 → x = 4。

If a fraction has a numerator that is an expression, like (x + 1)/3 = 2, treat the numerator as a grouped term. Multiply both sides by 3 to get x + 1 = 6, then x = 5.

如果分数的分子是一个表达式，比如 (x + 1)/3 = 2，就把分子看作一个整体。两边乘以3得到 x + 1 = 6，然后 x = 5。

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8. Check Your Solutions | 检验你的解

Substituting your answer back into the original equation is a crucial verification step. It confirms that both sides are equal and that no mistakes were made during solving.

把你的答案代回原方程是一个关键的检验步骤。它能够确认两边相等，并且求解过程中没有出现错误。

For the equation 2x + 5 = 13, if you found x = 4, check: LHS = 2(4) + 5 = 8 + 5 = 13, RHS = 13. They match, so the solution is correct.

对于方程 2x + 5 = 13，如果你算得 x = 4，检验：左边 = 2(4) + 5 = 8 + 5 = 13，右边 = 13。两边相等，因此解正确。

This habit also helps you spot errors like forgetting to apply an operation to both sides or misapplying the balance method. Checking takes only a few seconds but greatly improves accuracy.

这个习惯还能帮助你发现错误，比如忘记在两边都进行运算，或者错误地运用了天平法。检验只花几秒钟，但能大大提升准确性。

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9. Common Mistakes | 常见错误

Many mistakes come from rushing or not showing working. One typical error is applying an operation to only one side, which breaks the balance of the equation.

很多错误源于急于求成或不展示解题过程。一个典型的错误是只对一边进行运算，这就破坏了方程两边的平衡。

Another frequent mistake is mishandling negative signs when expanding brackets, for example writing 3 − 2(x + 1) = 3 − 2x + 1 instead of 3 − 2x − 2. Always use brackets carefully.

另一个常见错误是展开括号时符号处理不当，比如把 3 − 2(x + 1) 写成 3 − 2x + 1，而不是正确的 3 − 2x − 2。一定要谨慎使用括号。

Forgetting to check the solution or misreading the equation (e.g., mixing up x and a number) also leads to avoidable errors. Slow down, write each step, and double-check your work.

忘记检验解，或者看错方程（例如混淆x和数字），也会导致本可避免的错误。放慢速度，写下每一步，再复查一遍你的解答。

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10. Forming Equations from Words | 根据文字列方程

Many exam questions ask you to turn a written problem into an equation. Start by identifying the unknown and choose a letter to represent it. Then translate each phrase into a mathematical expression.

许多考试题目要求你将文字问题转化为方程。首先要找到未知量，并选择一个字母来表示它。然后把每一句描述翻译成数学表达式。

Example: “I think of a number, multiply it by 4, then add 7. The result is 31.” Let the number be n → 4n + 7 = 31. Solve: 4n = 24 → n = 6.

例子：“我心里想一个数，把它乘以4，再加上7，结果是31。”设这个数为n → 4n + 7 = 31。解：4n = 24 → n = 6。

Pay attention to words like “more than”, “less than”, “product”, and “sum”. They tell you which operations to use and sometimes the order in which they apply.

注意这里的用词，如“比……多”、“比……少”、“乘积”、“和”。它们会告诉你应该使用哪些运算，有时候还能体现运算的顺序。

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11. Real-Life Applications | 实际应用

Linear equations appear in everyday situations, such as calculating costs, distances, or comparing mobile phone plans. For instance, if a taxi charges a fixed flag-down fee plus a rate per kilometre, the total cost can be written as an equation.

线性方程出现在日常生活的各种场景中，比如计算费用、距离，或比较手机套餐。例如，如果出租车收一个固定的起步价加上每公里费用，总花费就可以写成一个方程。

If the flag-down fee is £3 and each kilometre costs £2, the equation for a journey of d kilometres with a total cost of £15 is 2d + 3 = 15. Solving gives d = 6 km.

如果起步价是3英镑，每公里2英镑，行驶d公里后总费用为15英镑的方程就是 2d + 3 = 15。解方程得到 d = 6 公里。

Using equations to model situations helps you make informed decisions, such as finding break-even points or comparing offers. This practical skill is built on the simple steps you practise in class.

用方程来建模能帮助你做出明智的决策，比如找到盈亏平衡点或比较不同优惠。这项实用技能正是建立在课堂中反复练习的那些简单步骤之上的。

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12. Summary and Practice Tips | 总结与练习建议

To solve any linear equation, remember the order: simplify both sides first (expand brackets, collect like terms), then use the balance method to isolate the variable, and finally check your answer.

解任何线性方程，都要记住这样的顺序：先化简两边（展开括号、合并同类项），然后用天平法分离出变量，最后检验你的答案。

Practice a mix of one-step, two-step, bracket and variable-on-both-sides equations daily. Write down each step clearly, even if you can do it mentally. This builds accuracy for exams.

每天混合练习一步方程、两步方程、带括号的方程和两边都有变量的方程。清晰地写出每一个步骤，即使你心算能做出来。这能锻炼你在考试中的准确性。

Use online quizzes, past paper questions, and challenge yourself with word problems. The key to confidence in algebra is consistent, step-by-step practice.

多做在线测验和历年真题，并用文字题挑战自己。学好代数的关键在于持之以恒、一步一个脚印地练习。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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