📚 Solving Linear Equations | 解一元一次方程
Linear equations are one of the most important building blocks in KS3 mathematics. They appear in many areas of the Cambridge curriculum, including algebra, number, geometry, and problem-solving. A linear equation is an equation where the unknown, usually written as x, is only raised to the power of 1. Learning to solve these equations confidently will help you with graphs, sequences, and real-life problems. This article explains the key ideas step by step, with bilingual explanations and examples to support your revision.
线性方程是 KS3 数学中最重要的基础模块之一。它们出现在剑桥课程的许多领域，包括代数、数、几何和问题解决。线性方程是指未知数（通常写成 x）只出现一次方的方程。学会自信地解这些方程将帮助你学习图像、数列和现实问题。本文逐步讲解核心概念，提供中英双语解释和例题，帮助你复习。
1. What is a Linear Equation? | 什么是线性方程？
0. These equations form a straight line when graphed, which is why they are called linear.
线性方程是表示两个表达式相等的数学语句。未知变量通常是 x，它的次方不超过 1，没有平方根，也没有变量的乘积。例如 x + 5 = 12、3x = 18 和 2x - 4 = 10。这些方程在图像上形成一条直线，因此称为线性方程。
In the Cambridge KS3 curriculum, you are expected to solve linear equations with integer, fraction, and decimal coefficients. You also need to understand the meaning of a solution. The solution is the value of x that makes the equation true when substituted back into the original equation.
在剑桥 KS3 课程中，你需要会解整数系数、分数系数和小数系数的线性方程。你还需要理解解的含义。解就是当把 x 的值代回原方程时，能使方程成立的数值。
2. The Balance Method | 天平法
The balance method is the foundation of solving linear equations. Imagine an old-fashioned balance scale. The left side and the right side must always have the same value. Whatever you do to one side, you must do to the other side, otherwise the balance is broken. This idea keeps the equation true.
天平法是解线性方程的基础。想象一个老式天平。左边和右边必须始终具有相同的值。你对一边做什么，就必须对另一边做同样的事情，否则天平就会被打破。这个思想使方程保持成立。
addition, subtraction, multiplication, and division. Using the balance method helps you avoid guessing and builds a strong algebraic foundation.
例如，如果你从 x + 3 = 10 的左边减去 3，你也必须从右边减去 3。这样得到 x = 7。同样的原理适用于加、减、乘、除。使用天平法可以帮助你避免猜测，并打下坚实的代数基础。
3. One-Step Equations | 一步方程
= 5, you add 3 to both sides. For multiplication and division, use the opposite operation.
一步方程可以通过一个逆运算来求解。如果方程是 x + 4 = 9，你在两边同时减去 4，使 x 单独出现。如果方程是 x - 3 = 5，你在两边同时加上 3。对于乘法和除法，使用相反的运算。
Here are the four basic inverse operations you must remember:
以下是你必须记住的四种基本逆运算：
Addition undoes subtraction | 加法抵消减法
Subtraction undoes addition | 减法抵消加法
Multiplication undoes division | 乘法抵消除法
Division undoes multiplication | 除法抵消乘法
For example, to solve 5x = 20, divide both sides by 5:
例如，解 5x = 20，两边同时除以 5：
5x ÷ 5 = 20 ÷ 5, so x = 4
Remember to line up your working vertically so each step is clear. This is especially important in exams where method marks are awarded.
记住把你的解题步骤垂直排列，使每一步都清晰。这在考试中尤其重要，因为方法分会被计入。
4. Two-Step Equations | 两步方程
sides to remove the constant term. Then divide both sides by 3 to find the value of x. Always reverse the order of operations: undo addition or subtraction before multiplication or division.
两步方程需要两个逆运算。一个典型例子是 3x + 2 = 14。首先，两边同时减去 2，去掉常数项。然后两边同时除以 3，求出 x 的值。始终要逆着运算顺序：先抵消加法或减法，再抵消乘法或除法。
Worked example:
解题示例：
3x + 2 = 14
3x = 14 - 2
3x = 12
x = 12 ÷ 3 = 4
You should write one operation per line. This makes your thinking clear and reduces mistakes. Many students try to do too much in their heads, especially when negative numbers are involved.
你应该每行只写一个运算。这会让你的思路清晰，减少错误。许多学生尤其在有负数时会试图用脑算太多步骤。
5. Equations with Brackets | 含括号的方程
butive law says a(b + c) = ab + ac, where a is multiplied by each term inside the bracket.
当方程含有括号时，你通常先用分配律展开括号。例如，2(x + 5) = 22 变成 2x + 10 = 22。然后作为两步方程求解。分配律是 a(b + c) = ab + ac，其中 a 要乘以括号内的每一项。
is is often faster, but only works when the outside number divides the right side exactly.
有时你可以先两边同时除以括号外的数。对于 2(x + 5) = 22，两边除以 2 得到 x + 5 = 11，所以 x = 6。这通常更快，但只有当括号外的数能整除右边时才适用。
gives 6x - 12 = 30, then 6x = 42, so x = 7. Always check your expansion before continuing.
对于更复杂的情况，先展开。例如：3(2x - 4) = 30 得到 6x - 12 = 30，然后 6x = 42，所以 x = 7。在继续之前始终检查你的展开是否正确。
6. Unknowns on Both Sides | 两边都有未知数
ect all x terms on one side and all constant terms on the other. Subtract the smaller x term from both sides to keep the coefficient positive.
有些方程在等号两边都有含 x 的项。例如，5x + 3 = 2x + 12。你的第一个目标是把所有含 x 的项移到一边，把所有常数项移到另一边。从两边减去较小的 x 项，使系数保持为正。
Worked example:
解题示例：
5x + 3 = 2x + 12
5x - 2x = 12 - 3
3x = 9
x = 3
Always move the variable terms to one side and the number terms to the other side using inverse operations. Some students try to move everything at once, which can lead to sign errors. Do it in clear, separate steps.
始终使用逆运算将变量项移到一边，将数字项移到另一边。有些学生试图一次性移动所有项，这会导致符号错误。要分清晰、独立的步骤进行。
7. Equations with Fraction Coefficients | 含分数系数的方程
multiply all terms by 3 to get x + 6 = 15. Then subtract 6 from both sides to find x = 9.
当线性方程含有分数时，最简单的方法通常是将每一项乘以最小公分母。例如，在 x/3 + 2 = 5 中，把每一项乘以 3，得到 x + 6 = 15。然后两边减去 6，得出 x = 9。
as two or more fractions, find the lowest common multiple of the denominators and multiply every term.
另一个例子是 (2x)/5 = 4。两边乘以 5 得到 2x = 20，然后除以 2 得到 x = 10。如果方程有两个或更多分数，求出分母的最小公倍数，并乘以每一项。
Multiplying by the lowest common denominator clears the fractions and often makes the equation much easier to solve. Be careful to multiply every term, including constant numbers on both sides of the equation.
乘以最小公分母可以清除分数，通常使方程更容易求解。注意要乘以每一项，包括方程两边的常数。
8. Writing Equations from Word Problems | 根据文字题列方程
Word problems require you to translate written information into an algebraic equation. Start by letting x represent the unknown quantity. Then identify how the other numbers relate to x. Words like 'sum', 'difference', 'product', and 'quotient' tell you which operation to use.
文字题要求你先把文字信息转化为代数方程。首先设 x 代表未知量。然后确定其他数字与 x 的关系。像“和”“差”“积”“商”这样的词语会告诉你使用哪种运算。
Example: "Three times a number plus 5 is 20." 0. Solve it to get x = 5. Then answer the question in words, not just with the value of x.
例如：“一个数的三倍加上 5 等于 20。”设这个数为 x，那么方程是 3x + 5 = 20。解方程得到 x = 5。然后用文字回答问题，而不只是给出 x 的值。
You should always read the question twice and underline key information. This helps you avoid confusing operations, such as using subtraction when the question says 'less than'. Practice writing equations before solving them.
你应该把题目读两遍，并在关键信息下划线。这有助于避免混淆运算，例如题目说“比……少”时你却用了减法。在解方程之前先练习列方程。
9. Checking Your Solution | 检验解
After you find a value for x, you should always substitute it back into the original equation. If the left side equals the right side, your solution is correct. This is called checking or verifying your answer. It is a quick way to catch simple arithmetic mistakes.
求出 x 的值后，你应该始终把它代回原方程。如果左边等于右边，你的解就是正确的。这叫做检验或验证答案。这是快速发现简单算术错误的好方法。
3 back: 4(3) - 1 = 12 - 1 = 11. The left side equals the right side, so x = 3 is correct.
例如：对于 4x - 1 = 11，我们得到 x = 3。把 3 代回去：4(3) - 1 = 12 - 1 = 11。左边等于右边，所以 x = 3 是正确的。
In exams, checking takes only a few seconds and can save you marks. If the check fails, go back through your working line by line. Do not just change the final answer without finding the mistake.
在考试中，检验只需要几秒钟，却可以帮你保住分数。如果检验不通过，就逐行回顾你的解题过程。不要只是更改最终答案而不找出错误。
10. Common Errors and How to Avoid Them | 常见错误及避免方法
Many KS3 students lose marks because of avoidable mistakes. One common error is only performing an operation on one side of the equation. Remember the balance method: if you add, subtract, multiply, or divide, you must do it to both sides.
许多 KS3 学生因可避免的错误而失分。一个常见错误是只在方程的一边进行运算。记住天平法：如果你加、减、乘或除，你必须对两边都做同样的运算。
6. Work slowly and write each step clearly, especially when negative numbers are involved.
另一个常见错误是负号处理不当。例如，在 x - 4 = -2 中，两边加上 4 得到 x = 2，而不是 x = -6。尤其在有负数时，要慢慢做并清晰写出每一步。
| Common error | 常见错误 | Correct approach | 正确做法 |
| Forgetting to operate on both sides | 忘记对两边进行运算 | Use the balance method every time | 每次都使用天平法 |
| Incorrect sign when moving terms | 移项时符号错误 | Use inverse operations carefully | 仔细使用逆运算 |
| Wrong order of inverse operations | 逆运算顺序错误 | Undo addition or subtraction first | 先抵消加法或减法 |
| Not checking the final answer | 不检验最终答案 | Substitute back into original equation | 代回原方程检验 |
Make a personal list of the mistakes you often make. Review this list before tests. Being aware of your common errors is one of the best ways to improve accuracy.
把你经常犯的错误列成一份个人清单。考试前复习这份清单。意识到自己的常见错误是提高准确率的最好方法之一。
11. Practice Strategy for Exams | 考试练习策略
To master linear equations, you need regular and focused practice. Start with one-step equations, then progress to two-step, brackets, unknowns on both sides, and fractions. Do not move on until you can solve each type confidently without help.
要掌握线性方程，你需要有规律、有重点地练习。从一步方程开始，然后逐步到两步方程、含括号的方程、两边有未知数的方程和分数方程。直到你能独立自信地解出每一类题，再进行下一步。
When practising, write out every step even if you think the problem is easy. This builds good habits for exams. Time yourself occasionally so you learn to work accurately under pressure. Mark your own work using a mark scheme, and understand where you lost method marks.
练习时写出每一步，即使你认为题目很简单。这会为考试养成好习惯。偶尔给自己计时，让自己学会在压力下准确答题。用评分标准批改自己的作业，并理解自己在哪里丢了方法分。
Cambridge KS3 papers often combine equations with geometry, such as angles in a triangle or perimeter problems. Try mixed questions that ask you to form and then solve an equation. This prepares you for real exam-style scenarios.
剑桥 KS3 试卷经常把方程与几何结合起来，例如三角形内角或周长问题。尝试一些要求你先列方程再求解的混合题。这会让你为真实考试题型做好准备。
Published by TutorHao | Cambridge Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图