📚 Solving Linear Equations: From One-Step to Word Problems | 解一元一次方程：从一步到文字题
Linear equations are the building blocks of algebra at KS3. They appear in nearly every Cambridge Lower Secondary Mathematics test, either as straightforward questions or hidden inside word problems. This article takes you through the full skill progression, from simple one-step equations to setting up equations from real-life situations.
一元一次方程是 KS3 代数的基础。它们几乎出现在每一份剑桥初中数学测试中，无论是直接考查还是隐藏在文字题里。本文将带你走完从简单一步方程到根据实际情境建立方程的完整技能进阶过程。
1. What Is a Linear Equation? | 什么是一元一次方程？
A linear equation is an algebraic statement that two expressions are equal. The unknown is usually written as a letter such as x, and the highest power of x is 1. You will never see x² or x³ in a linear equation.
一元一次方程是表示两个代数式相等的式子。未知数通常用 x 这样的字母表示，并且 x 的最高次数是 1。你永远不会在一元一次方程里看到 x² 或 x³。
tion because the unknown x is only raised to the power 1. The left side and right side are joined by an equals sign, which means they have the same value.
例如，2x + 3 = 11 是一个一元一次方程，因为未知数 x 的指数只是 1。等号连接左右两边，表示它们的值相等。
2. The Balance Method: Keeping Equations Equal | 天平法：保持等式平衡
Think of an equation as a balance scale. Whatever you do to one side of the equals sign, you must do exactly the same to the other side. This is called the balance method and it keeps the equation true.
把方程想象成一架天平。无论你对等号的一边做了什么，都必须对另一边做完全相同的操作。这叫做天平法，它能让方程始终保持成立。
If you subtract 4 from the left side, you must subtract 4 from the right side as well. If you divide the left side by 3, you must divide the right side by 3 too. This method gives you a safe way to isolate the unknown.
如果你从左边减去 4，就必须同时从右边减去 4。如果你把左边除以 3，就必须同时把右边也除以 3。这个方法为你提供了分离未知数的安全途径。
x + 4 = 10 → x + 4 − 4 = 10 − 4 → x = 6
3. Solving One-Step Equations | 解一步方程
In a one-step equation, only one operation separates you from the answer. You need to carry out the inverse operation to both sides to find the value of the unknown.
在一步方程中，只有一种运算挡在答案前面。你需要对两边同时进行逆运算，才能求出未知数的值。
tion is 5x = 35, divide both sides by 5 because division is the inverse of multiplication.
如果方程是 x + 7 = 15，就在两边同时减去 7，因为减法是加法的逆运算。如果方程是 5x = 35，就在两边同时除以 5，因为除法是乘法的逆运算。
ing the unknown mixed with a number, unless the question asks for an expression rather than an equation.
始终把最终答案写成简洁的形式，例如 x = 8。除非题目要求的是代数式而不是方程，否则不要把未知数和数字混在一起。
4. Solving Two-Step Equations | 解两步方程
A two-step equation contains two operations, such as multiplication and addition. To solve it, reverse the operations in the correct order: first undo addition or subtraction, then undo multiplication or division.
两步方程包含两种运算，例如乘法和加法。解这类方程时，要按正确的顺序逆运算：先处理加减，再处理乘除。
vide both sides by 2 to get x = 6. Reversing the order would make the working much harder.
以方程 2x + 5 = 17 为例。先在两边同时减去 5，得到 2x = 12。然后再把两边同时除以 2，得到 x = 6。如果颠倒顺序，计算会困难很多。
2x + 5 = 17 → 2x = 12 → x = 6
5. Equations with Brackets | 带括号的方程
When an equation has brackets, expand them first using the distributive law. Multiply the term outside the bracket by every term inside the bracket before you start using the balance method.
当方程带有括号时，要先用分配律展开括号。先让括号外的项与括号内的每一项相乘，然后再开始使用天平法。
expansion. Then subtract 6 from both sides to get 3x = 15, and divide by 3 to find x = 5.
例如，3(x + 2) = 21 在展开后变成 3x + 6 = 21。然后两边同时减去 6 得到 3x = 15，再除以 3 求出 x = 5。
If there is a negative sign before the bracket, treat it as multiplying by −1. So −(x − 4) becomes −x + 4, not −x − 4.
如果括号前有负号，就把负号看成乘以 −1。因此 −(x − 4) 展开后应为 −x + 4，而不是 −x − 4。
6. Unknowns on Both Sides | 两边都有未知数的方程
Some equations have the unknown appearing on both sides of the equals sign. Start by collecting all x terms on one side and all constant terms on the other side.
有些方程的两边都会出现未知数。解题时先把所有含 x 的项移到一边，把所有常数项移到另一边。
h sides to get 2x + 2 = 10. Then subtract 2 from both sides and divide by 2 to find x = 4.
对于 5x + 2 = 3x + 10，先在两边同时减去 3x，得到 2x + 2 = 10。然后两边同时减去 2，再除以 2，求出 x = 4。
It is usually easier to eliminate the smaller x term first, as this keeps the coefficient positive. This reduces the chance of sign errors later in your working.
通常先消去较小的 x 项会更容易，因为这样系数可以保持为正数。这能减少后续计算中出现符号错误的可能性。
7. Equations with Fractions and Decimals | 分数与小数方程
Fractions and decimals often make equations look harder than they really are. The key is to clear the fraction or decimal early, using multiplication by a suitable power of 10 or by the denominator.
分数和小数常常让方程看起来比实际更难。关键在于尽早消去分数或小数，可以用适当的 10 的幂或分母去乘。
x + 12 = 28. Then subtract 12 to find x = 16. This removes the fraction in one clean step.
对于 x/4 + 3 = 7，把每一项都乘以 4，得到 x + 12 = 28。再减去 12 求出 x = 16。这样一步就能把分数去掉。
2 to get x = 6. Turning decimals into whole numbers makes the arithmetic much friendlier.
如果你看到 0.5x = 3，就把两边同时乘以 2，得到 x = 6。把小数变成整数会让计算更加简单。
8. Checking Your Solution by Substitution | 代入法检验答案
After finding a solution, always check it by substituting the value back into the original equation. Both sides should give the same number if your answer is correct.
求出解之后，一定要把答案代回原方程进行检验。如果答案正确，等号两边应该得到相同的数。
back: 2(6) + 5 = 12 + 5 = 17. Since the right side is also 17, the solution is confirmed.
如果你解出 2x + 5 = 17 并且得到 x = 6，就把它代回去：2(6) + 5 = 12 + 5 = 17。因为右边也是 17，所以答案得到确认。
Checking is especially important in exams because it can catch small arithmetic mistakes before you move on. It only takes a few seconds and can save several marks.
在考试中检验尤其重要，因为它能在你继续做下一题之前发现小的计算错误。检验只需要几秒钟，却可能保住好几分。
9. Setting Up Equations from Word Problems | 文字题建立方程
Word problems ask you to translate everyday language into algebra. Read the problem carefully, identify the unknown quantity, and give it a letter such as x. Then build an equation using the relationships described.
文字题要求你把日常语言翻译成代数。仔细读题，找出未知量，并用 x 这样的字母表示它。然后根据题目描述的关系建立方程。
For example, "I think of a number, double it and add 7. The result is 23." Let the number be x, so the equation is 2x + 7 = 23. Solving gives x = 8.
例如，“我想了一个数，把它乘以 2 再加 7，结果是 23。” 设这个数为 x，那么方程就是 2x + 7 = 23。解方程得到 x = 8。
Practise spotting key words: "sum" means addition, "product" means multiplication, "less than" means subtraction, and "twice" means multiplication by 2.
练习辨认关键词：“和”表示加法，“积”表示乘法，“比……少”表示减法，“两倍”表示乘以 2。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One common mistake is forgetting to do the same operation on both sides of the equation. This breaks the balance rule and gives an incorrect answer.
一个常见错误是忘记在等号两边做同样的运算。这会破坏天平规则，导致答案错误。
Another frequent error is mishandling negative signs, especially when expanding brackets or moving terms. Write each step neatly and double-check that −(x − 3) becomes −x + 3.
另一个常见错误是处理负号时出错，尤其是在展开括号或移项时。每一步都要写清楚，并且反复检查 −(x − 3) 展开后应为 −x + 3。
Finally, do not combine unlike terms. For example, 3x + 4 cannot be simplified to 7x because one term has an unknown and the other does not.
最后，不要合并不同类项。例如，3x + 4 不能化简成 7x，因为一个含未知数，另一个不含未知数。
11. Exam-Style Practice Questions | 考试风格练习题
Try these short questions to test your skills. Set out each step clearly and check your final answer by substitution.
试着做下面这些简短题目来检验你的技能。每一步都要写清楚，并用代入法检验最终答案。
4x − 3 = 17
5(x + 2) = 35
6x + 1 = 2x + 17
x/5 + 2 = 9
t x = 5. For the second, expand to 5x + 10 = 35, subtract 10 and divide by 5 to get x = 5.
第一题先在两边加 3，再除以 4，得到 x = 5。第二题先展开为 5x + 10 = 35，再减去 10 并除以 5，得到 x = 5。
x and 1, so x = 4. The fourth gives x/5 = 7, and multiplying both sides by 5 gives x = 35.
第三题在减去 2x 和 1 后得到 4x = 16，所以 x = 4。第四题得到 x/5 = 7，两边同时乘以 5 后得到 x = 35。
12. Key Takeaways for KS3 | KS3 关键要点
To master linear equations, you need a strong command of inverse operations and the balance method. Always keep the equation balanced by doing the same thing to both sides.
要掌握一元一次方程，你需要熟练运用逆运算和天平法。始终对两边做相同的运算，保持方程平衡。
Move from simple examples to harder ones step by step: one-step, two-step, brackets, unknowns on both sides, and finally word problems. Regular practice builds speed and confidence.
从简单例子逐步过渡到较难的题目：一步、两步、带括号、两边有未知数，最后是文字题。经常练习能提高速度和信心。
Before any class test or checkpoint, review your written solutions and look for repeated mistakes. A few minutes of careful checking can make a real difference to your final score.
在课堂测验或 checkpoint 之前，复习你的书面解题过程并找出重复出现的错误。认真检查几分钟就能对最终成绩产生真正的影响。
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