📚 Mastering Linear Equations | 掌握一元一次方程
s and a ≠ 0. In a linear equation, the highest power of the variable x is 1, so there are no x², x³, or fractional powers. Solving a linear equation means finding the value of x that makes the equation true. This is one of the most important skills in the Cambridge KS3 Mathematics course because it appears in many problem-solving contexts, from number puzzles to geometry and data questions.
一元一次方程是可以写成 ax + b = c 形式的方程，其中 a、b、c 是数且 a ≠ 0。在一元一次方程中，未知数 x 的最高次数为 1，因此不会出现 x²、x³ 或分数次幂。解一元一次方程就是求出使等式成立的 x 值。这是剑桥 KS3 数学课程中最重要的技能之一，因为它出现在许多问题解决情境中，从数字谜题到几何和数据问题都有涉及。
ide equals the right-hand side. If x = 3, then 3(3) + 4 = 9 + 4 = 13, so x = 3 is correct.
例如，3x + 4 = 13 是一元一次方程，因为未知数 x 只以一次幂出现。要判断一个数是否为方程的解，可以把它代入方程，检查左边是否等于右边。如果 x = 3，则 3(3) + 4 = 9 + 4 = 13，所以 x = 3 是正确的解。
1. What is a linear equation? | 什么是一元一次方程？
s not zero. The variable, usually written as x or n, only appears to the power 1. The word 'linear' , you will mostly work with equations such as 2x + 5 = 15, 7m - 3 = 18, and 4(x + 2) = 24.
一元一次方程总可以化简为 ax + b = c 的形式，其中 a、b、c 是已知数且 a 不为零。变量通常写作 x 或 n，只以一次幂出现。“一次”这个词来源于相关表达式的图像是一条直线。在 KS3 阶段，你主要会接触到像 2x + 5 = 15、7m - 3 = 18 和 4(x + 2) = 24 这样的方程。
The key idea is that the equals sign acts as a balance point. Whatever operation you perform on one side, you must perform the same operation on the other side. This balance rule is the foundation of all equation solving. Without it, the equation would no longer be true.
核心思想是等号相当于一个平衡点。无论你对一边做什么运算，都必须对另一边做同样的运算。这种平衡规则是解所有方程的基础。如果不遵守这个规则，方程就不再成立。
2. Key vocabulary and notation | 关键术语与符号
the coefficient, x is the variable, 2 and 17 are constants, and the whole expression 5x + 2 is the left-hand side while 17 is the right-hand side. The equals sign separates the two sides.
在解方程之前，了解一些专门术语会很有帮助。在方程 5x + 2 = 17 中，数字 5 叫做系数，x 是变量，2 和 17 是常数，整个表达式 5x + 2 是左边，17 是右边。等号把左右两边分开。
- Variable | 变量：a letter that stands for an unknown number, usually x, y, n or m.
- Coefficient | 系数：the number multiplying the variable, such as 5 in 5x.
- Constant | 常数：a fixed number, such as 2 or 17.
- Left-hand side (LHS) | 左边：everything before the equals sign.
- Right-hand side (RHS) | 右边：everything after the equals sign.
- Solution | 解：the value of the variable that makes the equation true.
Using precise language helps you follow the steps and communicate your method clearly in tests. For example, saying 'subtract 2 from both sides' is clearer than saying 'take 2 over to the other side', because the second phrase can lead to sign mistakes.
使用准确的语言有助于你理解步骤并在考试中清晰地表达方法。例如，说“两边同时减去 2”比说“把 2 移到另一边”更清楚，因为后一种说法容易导致符号错误。
3. Solving by inverse operations | 逆运算解法
The simplest linear equations can be solved by reversing the operations that have been applied to the variable. If the equation involves addition, subtract the same number from both sides. If it involves subtraction, add the same number to both sides. If it involves multiplication, divide both sides by the coefficient. If it involves division, multiply both sides.
最简单的一元一次方程可以通过逆运算来解。如果方程含有加法，就在两边同时减去同一个数；如果含有减法，就在两边同时加上同一个数；如果含有乘法，就在两边同时除以系数；如果含有除法，就在两边同时乘以同一个数。
x + 5 = 12
x = 12 - 5
x = 7
3x = 18
x = 18 ÷ 3
x = 6
e is to divide before subtracting, so always deal with addition or subtraction before multiplication or division.
当方程含有两种运算时，例如 2x + 5 = 13，要按正确顺序逆向运算。先两边减去 5 得 2x = 8，再两边除以 2 得 x = 4。一个常见错误是先除后减，所以一定要先处理加减，再处理乘除。
4. Equations with brackets | 含括号的方程
brackets are removed, the equation can be solved using inverse operations. Expanding brackets correctly is essential before moving terms around.
如果方程含有括号，应先用分配律去括号。例如，2(x + 3) 表示 2 × x 加上 2 × 3，得到 2x + 6。去掉括号后，就可以用逆运算解方程。正确去括号是移项之前的关键步骤。
2(x + 3) = 14
2x + 6 = 14
2x = 14 - 6
2x = 8
x = 8 ÷ 2
x = 4
t x = -1. Many students forget that the minus sign changes both terms inside the brackets.
注意括号外的负号。在方程 -3(x - 2) = 9 中，要把 x 和 -2 都乘以 -3：-3 × x = -3x，-3 × -2 = +6，得到 -3x + 6 = 9。然后两边减 6 得 -3x = 3，再除以 -3 得 x = -1。很多学生会忘记负号会改变括号内两项的符号。
5. Unknowns on both sides | 两边均含未知数
When an unknown appears on both sides of the equation, collect the x terms on one side and the constants on the other side. This is usually done by adding or subtracting the same x term from both sides. Choose the side that avoids negative coefficients if possible, but either side will work if the operations are performed correctly.
当未知数同时出现在方程两边时，把含 x 的项移到一边，常数移到另一边。通常通过在两边加减同一个含 x 的项来实现。如果可能，尽量选择能避免负系数的一边，但只要运算正确，选哪一边都可以。
5x - 3 = 2x + 6
5x - 3 - 2x = 2x + 6 - 2x
3x - 3 = 6
3x = 6 + 3
3x = 9
x = 9 ÷ 3
x = 3
After collecting the x terms, solve the equation as before. It is important to write each step on a new line and to make sure the equation remains balanced. If you subtract 2x from the left, you must also subtract 2x from the right. This method works for any linear equation with unknowns on both sides.
合并含 x 的项后，像之前一样解方程。每一步写在新的一行，并确保方程保持平衡。如果在左边减去 2x，右边也必须减去 2x。这种方法适用于任何两边含未知数的一元一次方程。
6. Equations with fractions | 含分数的方程
reduces mistakes. For example, in x/2 + 3 = 7, multiply everything by 2 to get x + 6 = 14.
含分数的方程可以通过把每一项都乘以最小公分母来解。这样可以消去分母，得到一个更简单的方程。当然，你也可以直接使用逆运算，但先消去分数通常能减少错误。例如，在 x/2 + 3 = 7 中，全式乘以 2 得到 x + 6 = 14。
x/2 + 3 = 7
2(x/2) + 2(3) = 2(7)
x + 6 = 14
x = 14 - 6
x = 8
multiply the whole of each term, not just part of it, otherwise the equation becomes unbalanced.
当分数带有负号时，要小心符号与分子或分母一起保留。对于 x/3 - 2 = 4，把每一项都乘以 3 得到 x - 6 = 12，然后两边加 6 得 x = 18。一定要对每一项的整体进行乘法，不能只乘其中的一部分，否则方程会失去平衡。
7. Checking your solution | 检验解
After finding a solution, always substitute it back into the original equation to check that it works. This simple habit can catch many small mistakes, such as an incorrect sign or a missed operation. If the left-hand side does not equal the right-hand side, go back and check your working line by line.
求出解后，一定要把它代回原方程检验是否正确。这个简单的习惯可以发现许多小错误，比如符号错误或漏掉某步运算。如果左边不等于右边，就要逐行回去检查计算过程。
Check x = 8 in x/2 + 3 = 7:
8/2 + 3 = 4 + 3 = 7
7 = 7 ✔
In an exam, checking does not take long and can save valuable marks. Even if you are confident, a quick substitution confirms that your answer satisfies the original conditions. For word problems, the check also makes sure the answer is reasonable in the context of the question.
在考试中，检验花不了多少时间，却能保住宝贵的分数。即使你很有信心，快速代入也能确认答案满足原始条件。对于文字题，检验还可以确保答案在题目情境中是合理的。
8. Word problems and forming equations | 文字题与列方程
Many KS3 exam questions ask you to form an equation from a word problem before solving it. Start by reading the problem twice and underline the key numbers and relationships. Choose a letter, such as x or n, to represent the unknown quantity, then translate each part of the sentence into algebraic symbols. Finally, solve the equation and answer the original question in words.
许多 KS3 考试题会要求你先根据文字题列出方程，然后求解。先仔细读题两遍，划出关键数字和关系。选择一个字母，如 x 或 n，表示未知量，然后把句子的每一部分翻译成代数符号。最后解方程，并用文字回答原题。
For example, 'three times a number plus five equals twenty' can be translated as 3n + 5 = 20. Solving gives n = 5. Another example: 'the sum of a number and seven is twice the number' rect equation is often the hardest part, so practise turning words into symbols regularly.
例如，“一个数的三倍加五等于二十”可以翻译为 3n + 5 = 20，解得 n = 5。另一个例子：“一个数与七的和是这个数的两倍”可写成 n + 7 = 2n，解得 n = 7。列对方程往往是最难的部分，所以要多练习把文字转化为符号。
9. Common errors and how to avoid them | 常见错误及避免方法
ting 5 from both sides. Another mistake is forgetting to distribute a negative sign to every term inside brackets, which changes the equation completely.
学生在解一元一次方程时常犯同样的错误。一个常见错误是只对一边运算，比如从 x + 5 = 12 直接写成 x = 12，而不是两边都减 5。另一个错误是忘记把负号分配给括号内的每一项，这会完全改变方程。
| Common error | 常见错误 | Correct approach | 正确做法 |
| Only changing one side | 只改变一边 | Apply the same operation to both sides | 对两边做同样运算 |
| Adding instead of subtracting | 该减却加 | Use inverse operations carefully | 仔细使用逆运算 |
| Mis-handling negative brackets | 负括号处理错误 | Multiply every term inside by the sign outside | 括号内每一项都乘以外面的符号 |
To avoid these errors, write each step neatly on a separate line and check the balance after every operation. If you are unsure, substitute a simple number into the original equation to see whether your working makes sense. Regular practice with small, careful steps is the best way to improve accuracy.
为避免这些错误，每一步都整齐地写在新的一行上，并在每次运算后检查平衡。如果不确定，可以代入一个简单数字到原方程中，看看计算是否合理。通过小步骤、细致的反复练习是提高准确率的最好方法。
10. Exam-style practice and summary | 考试风格练习与总结
get 2x - 8 = 8, add 8 to both sides to get 2x = 16, and finally divide by 2 to get x = 8.
为了准备剑桥 KS3 考试，要混合练习各种类型的方程。例如，解 4(x - 2) = 2x + 8。先展开左边得 4x - 8 = 2x + 8，然后两边减 2x 得 2x - 8 = 8，两边加 8 得 2x = 16，最后除以 2 得 x = 8。
4(x - 2) = 2x + 8
4x - 8 = 2x + 8
4x - 8 - 2x = 2x + 8 - 2x
2x - 8 = 8
2x = 8 + 8
2x = 16
x = 16 ÷ 2
x = 8
In summary, always keep the equation balanced, reverse operations in the correct order, expand brackets before moving terms, and check your final answer by substitution. These habits will help you solve nearly any linear equation you meet at KS3 level, and they will provide a strong foundation for IGCSE work later on.
总而言之，要始终保持方程平衡，按正确顺序逆向运算，先展开括号再移项，最后用代入法检验答案。这些习惯将帮助你解决 KS3 阶段几乎所有的线性方程，并为今后的 IGCSE 学习打下坚实基础。
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