📚 Solving Linear Equations: From Balancing to Word Problems | 解一元一次方程：从平衡法到文字题
Linear equations are one of the most important tools in KS3 mathematics. They describe a rule where the unknown value, usually written as x, is multiplied, divided, added to or subtracted from numbers. Being able to solve these equations builds a bridge between arithmetic and algebra, and it prepares you for graphs, formulas and real-world problems.
一元一次方程是 KS3 数学中最重要的工具之一。它们描述了一条规则：未知数通常写成 x，并与数字相乘、相除、相加或相减。能够解这些方程就在算术和代数之间架起了一座桥梁，并为图像、公式和实际问题做好准备。
1. What Is a Linear Equation? | 什么是一元一次方程？
A linear equation in one variable has an unknown, often x, and no powers higher than 1. For example, 2x + 3 = 11 is linear because x is only raised to the first power.
一元一次方程有一个未知数，通常用 x 表示，并且未知数的最高次数是 1。例如 2x + 3 = 11 就是一次方程，因为 x 只有一次方。
The word “linear” comes from the fact that such an equation can be represented by a straight line on a graph. When you solve it algebraically, you are finding the x-coordinate where two lines would meet.
“linear” 一词源于这种方程在图像上可以表示为一条直线。当你用代数方法解方程时，你实际上是在求两条直线交点的 x 坐标。
- Examples of linear equations: x + 5 = 12, 3y − 4 = 8, 7 = 2n + 1
- Non-examples: x² + 2 = 6, y³ − 5 = 1, because the power is greater than 1.
- 一次方程示例：x + 5 = 12、3y − 4 = 8、7 = 2n + 1
- 非一次方程示例：x² + 2 = 6、y³ − 5 = 1，因为未知数的次数大于 1。
2. The Balance Method | 平衡法
To solve an equation, imagine a balance scale. Whatever you do to one side, you must do to the other side to keep it balanced. This is the golden rule of algebra.
解方程时，想象一台天平。无论你对一边做了什么，都必须对另一边做同样的事情，才能保持平衡。这是代数的黄金法则。
For example, if you subtract 5 from the left side of x + 5 = 12, you must also subtract 5 from the right side. This gives x = 7 without changing the equality.
例如，如果你从 x + 5 = 12 的左边减去 5，你也必须从右边减去 5。这样得到 x = 7，而且等式仍然成立。
Keeping the balance method in mind helps you avoid the common mistake of only changing one side of the equation.
牢记平衡法可以帮助你避免只改变方程一边的常见错误。
3. Solving One-Step Equations | 解一步方程
One-step equations need only one inverse operation. To undo addition, subtract the same number from both sides. To undo subtraction, add the same number to both sides.
一步方程只需要一次逆运算。要撤销加法，就两边都减去同一个数；要撤销减法，就两边都加上同一个数。
For multiplication and division, use the opposite operation. If the variable is multiplied by a number, divide both sides by that number. If the variable is divided by a number, multiply both sides by that number.
对于乘法和除法，要使用相反的运算。如果变量乘以某个数，就两边都除以这个数；如果变量除以某个数，就两边都乘以这个数。
x − 4 = 9 → x − 4 + 4 = 9 + 4 → x = 13
5x = 35 → 5x ÷ 5 = 35 ÷ 5 → x = 7
4. Solving Two-Step Equations | 解两步方程
A two-step equation involves two operations. You must undo them in reverse order, starting with the operation furthest from the variable. Usually this means you deal with addition or subtraction first, then multiplication or division.
两步方程包含两种运算。你必须按照与运算顺序相反的顺序来撤销它们，从离变量最远的运算开始。通常这意味着先处理加法或减法，再处理乘法或除法。
Take the equation 2x + 5 = 17. The unknown x is first multiplied by 2, then 5 is added. To solve, subtract 5 from both sides, then divide both sides by 2.
以方程 2x + 5 = 17 为例。未知数 x 先乘以 2，然后再加上 5。要解它，先两边都减去 5，再两边都除以 2。
2x + 5 = 17 → 2x = 12 → x = 6
Always write each step on a new line and keep the equals signs aligned. This makes your working clear and easy to check.
每一步都要另起一行写，并保持等号对齐。这样你的解题过程会很清楚，也便于检查。
5. Equations with Brackets | 带括号的方程
When an equation has brackets, expand them first using the distributive law: a(b + c) = ab + ac. This turns the expression into a simpler form without brackets.
当方程含有括号时，先用分配律展开括号：a(b + c) = ab + ac。这样可以把表达式变成不含括号的简单形式。
For example, 3(x + 4) = 27 becomes 3x + 12 = 27. Then subtract 12 from both sides and divide by 3.
例如，3(x + 4) = 27 变成 3x + 12 = 27。然后两边都减去 12，再除以 3。
3(x + 4) = 27 → 3x + 12 = 27 → 3x = 15 → x = 5
Be careful with negative signs: −2(x − 3) = −2x + 6. A negative outside a bracket changes the sign of every term inside.
注意负号：−2(x − 3) = −2x + 6。括号外的负号会改变括号内每一项的符号。
6. Equations with Unknowns on Both Sides | 未知数在方程两边
If the unknown appears on both sides, collect the variable terms on one side and the number terms on the other. You can do this by adding or subtracting the same term from both sides.
如果未知数出现在方程的两边，就把含变量的项移到一边，把数字项移到另一边。你可以通过两边同时加上或减去同一个项来实现。
Solve 5x − 2 = 2x + 10. First subtract 2x from both sides to get 3x − 2 = 10. Then add 2 to both sides and divide by 3.
解方程 5x − 2 = 2x + 10。先两边都减去 2x，得到 3x − 2 = 10。然后两边都加上 2，再除以 3。
5x − 2 = 2x + 10 → 3x − 2 = 10 → 3x = 12 → x = 4
It does not matter which side you collect the variable on, as long as you keep the balance. Choosing the side with the larger coefficient can reduce negative signs.
你把变量移到哪一边并不重要，只要保持平衡即可。选择系数较大的一边，往往可以减少负号的出现。
7. Fractional Coefficients | 分数系数
When an equation contains fractions, you can multiply both sides by the denominator to clear the fraction. This often makes the equation easier to solve.
当方程含有分数时，你可以两边都乘以分母来去掉分数。这通常会让方程更容易解。
Solve x/4 + 2 = 5. First subtract 2 to get x/4 = 3. Then multiply both sides by 4.
解方程 x/4 + 2 = 5。先两边都减去 2，得到 x/4 = 3。然后两边都乘以 4。
x/4 + 2 = 5 → x/4 = 3 → x = 12
If there are two or more fractions, multiply every term by the lowest common denominator. For example, x/2 + x/3 = 5 becomes 3x + 2x = 30 after multiplying by 6.
如果有两个或更多分数，就把每一项都乘以最小公分母。例如，x/2 + x/3 = 5 在乘以 6 后变成 3x + 2x = 30。
8. Forming Equations from Word Problems | 根据文字题列方程
Many problems ask you to form an equation before solving it. Read the problem carefully, let the unknown be a letter, then translate the words into algebra step by step.
许多题目要求你先列出方程再求解。仔细读题，用字母表示未知数，然后把文字一步一步翻译成代数语言。
Example: “I think of a number, double it and add 7. The result is 19.” Let n be the number. The equation is 2n + 7 = 19.
例题：”我想一个数，把它乘以 2，再加上 7，结果是 19。” 设这个数为 n，方程是 2n + 7 = 19。
2n + 7 = 19 → 2n = 12 → n = 6
Always check that your answer makes sense in the original wording, not just in the equation you wrote.
一定要检查你的答案在原来的文字题中是否合理，而不仅仅是在你列出的方程中合理。
9. Checking Solutions | 检验解
After you find a solution, substitute it back into the original equation. Both sides should give the same value.
求出解之后，把解代入原方程。两边应该得到相同的值。
For 2x + 5 = 17, we found x = 6. Check: 2 × 6 + 5 = 12 + 5 = 17, which matches the right side.
对于 2x + 5 = 17，我们得到 x = 6。检验：2 × 6 + 5 = 12 + 5 = 17，与右边一致。
This habit catches arithmetic errors and sign mistakes before they affect your final answer. It is especially useful in tests.
这个习惯可以在算术错误和符号错误影响最终答案之前发现它们。在考试中尤其有用。
10. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Students often forget to do the same operation on both sides. This breaks the balance and leads to the wrong answer.
学生经常忘记在两边做同样的运算。这会破坏平衡，导致答案错误。
Another common mistake is mishandling negative signs, especially when expanding brackets like −2(x − 3). Remember that −2 × −3 = +6.
另一个常见错误是负号处理不当，特别是在展开像 −2(x − 3) 这样的括号时。记住 −2 × −3 = +6。
- Forgetting to reverse the order of operations in two-step equations.
- Combining unlike terms, such as writing 2x + 3 = 5x.
- Dividing incorrectly when the coefficient is negative.
- 在两步方程中忘记颠倒运算顺序。
- 合并不同类项，例如写出 2x + 3 = 5x。
- 当系数为负数时除法出错。
To avoid these errors, write every step clearly, use the balance method, and always check your final solution.
为了避免这些错误，每一步都要写得清楚，使用平衡法，并且始终检验你的最终解。
11. Practice Questions and Answers | 练习题与答案
Try these questions before looking at the answers. Solve each equation for the unknown.
在看答案之前先试做这些题。解出每个方程中的未知数。
| Question | 题目 | Answer | 答案 |
|---|---|
| x + 9 = 14 | x = 5 |
| 4x − 3 = 13 | x = 4 |
| 3(x − 2) = 18 | x = 8 |
| 7x + 1 = 3x + 17 | x = 4 |
| x/5 + 3 = 7 | x = 20 |
For each answer, substitute it back into the original equation to check that both sides are equal.
对于每个答案，把它代入原方程，检查两边是否相等。
12. Summary | 总结
Linear equations are solved by using inverse operations and keeping both sides balanced. Start by simplifying brackets and clearing fractions, then collect like terms, and finally isolate the unknown.
解一元一次方程需要使用逆运算并保持两边平衡。先化简括号和清除分数，然后合并同类项，最后把未知数单独留在一边。
Remember the golden rule: whatever you do to one side, do to the other. Check your answer by substitution. These skills are the foundation for much of secondary mathematics.
记住黄金法则：你对一边做了什么，就一定要对另一边做同样的事情。通过代入法检查答案。这些技能是中学数学许多内容的基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导