📚 Solving Linear Equations | 解一元一次方程
Linear equations are the foundation of algebra. They help us represent real-life situations using symbols and find unknown values. In Key Stage 3, mastering how to solve these equations builds confidence for all future maths topics, from graphs to problem-solving. This article breaks down every essential method you need, with clear examples and common pitfalls to avoid.
线性方程是代数的基础。它们帮助我们使用符号表示现实生活中的情境，并求出未知量。在 KS3 阶段，掌握如何解这些方程能够为你日后学习从图像到解决问题的所有数学主题建立信心。本文分解了你需要掌握的每一种基本方法，并配有清晰的示例和需要避免的常见错误。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is an algebraic statement where the highest power of the variable is 1. It has an equals sign and usually looks like ax + b = c or ax + b = cx + d. The goal is to find the value of the unknown that makes the equation true.
线性方程是一个代数陈述，其中变量的最高次幂为 1。它有一个等号，通常形如 ax + b = c 或 ax + b = cx + d。我们的目标是求出使方程成立的未知数的值。
For example, in the equation x + 5 = 12 st be 7 because 7 + 5 = 12. That value is called the solution or the root of the equation.
例如，在方程 x + 5 = 12 中，未知数 x 必须是 7，因为 7 + 5 = 12。这个值叫做方程的解或方程的根。
Linear equations can have one solution, no solution, or infinitely many solutions, but at KS3 we focus on those with exactly one sensible answer.
线性方程可以有一个解、没有解或有无穷多个解，但在 KS3 阶段我们主要关注那些恰好有一个合理答案的方程。
2. The Balancing 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 is the golden rule of solving equations.
把方程想象成一个平衡的天平。你对一边所做的任何操作，必须对另一边做完全相同的操作，以保持天平平衡。这是解方程的黄金法则。
If we have x + 3 = 10, we subtract 3 from both sides: x + 3 − 3 = 10 − 3, which simplifies to x = 7. The balance is preserved.
如果我们有 x + 3 = 10，那么在两边同时减去 3：x + 3 − 3 = 10 − 3，化简得到 x = 7。天平得以保持平衡。
This idea works for multiplication and division too. For 4x = 20, divide both sides by 4: 4x ÷ 4 = 20 ÷ 4, giving x = 5.
这个思想同样适用于乘法和除法。对于 4x = 20，两边同时除以 4：4x ÷ 4 = 20 ÷ 4，得到 x = 5。
3. Using Inverse Operations | 使用逆运算
To isolate the variable, we undo operations by applying their inverses. Addition and subtraction are inverse operations; multiplication and division are inverse operations.
为了把变量单独解出来，我们通过施加逆运算来撤销原有的运算。加法和减法互为逆运算；乘法和除法互为逆运算。
For example, solve 2x + 5 = 17. First, subtract 5 (the inverse of +5) from both sides to get 2x = 12. Then divide by 2 (the inverse of multiplication by 2) to get x = 6.
例如，解方程 2x + 5 = 17。首先，两边同时减去 5（+5 的逆运算），得到 2x = 12。然后两边同时除以 2（乘以 2 的逆运算），得到 x = 6。
Always perform addition/subtraction steps before multiplication/division steps unless brackets tell you otherwise. Reversing BIDMAS in reverse order: undo addition/subtraction first, then undo multiplication/division.
除非括号有不同指示，否则始终先执行加/减步骤，再执行乘/除步骤。也就是逆着 BIDMAS 顺序：先撤销加减，再撤销乘除。
4. Equations with Brackets | 带括号的方程
When an equation contains brackets, expand them first using the distributive law. For instance, 3(x + 4) = 21 expands to 3x + 12 = 21.
当方程含有括号时，首先使用分配律展开括号。例如，3(x + 4) = 21 展开后变成 3x + 12 = 21。
After expanding, solve as normal: subtract 12 from both sides to get 3x = 9, then divide by 3 to find x = 3.
展开之后，按常规方法求解：两边同时减去 12，得到 3x = 9，然后两边同时除以 3，求出 x = 3。
Sometimes you might see a negative sign before a bracket, such as 5 − 2(x − 1) = 3. Be careful: the negative sign flips the signs inside. It becomes 5 − 2x + 2 = 3, then 7 − 2x = 3.
有时你可能会看到括号前有一个负号，比如 5 − 2(x − 1) = 3。要注意：负号会改变括号内各项的符号。它变成 5 − 2x + 2 = 3，然后化为 7 − 2x = 3。
5. Unknowns on Both Sides | 未知数在等号两边
Equations like 5x + 2 = 3x + 10 have x on both sides of the equals sign. The strategy is to collect all variable terms on one side and all constant terms on the other.
像 5x + 2 = 3x + 10 这样的方程在等号两边都有 x。解题策略是把所有的变量项集中到一边，把所有的常数项集中到另一边。
Subtract the smaller variable term from both sides to keep things positive. Here, subtract 3x: 5x − 3x + 2 = 3x − 3x + 10, giving 2x + 2 = 10. Then proceed normally, subtracting 2 and dividing by 2 to get x = 4.
从两边同时减去较小的变量项，以保持系数为正。这里，减去 3x：5x − 3x + 2 = 3x − 3x + 10，得到 2x + 2 = 10。然后按常规步骤，先减 2 再除以 2，得到 x = 4。
If the variable ends up with a negative coefficient, it is perfectly okay to multiply both sides by −1 at the end to make the solution positive. For example, −x = 5 becomes x = −5.
如果最终变量的系数是负的，在最后把两边同乘以 −1 使其变为正数是完全可以的。例如，−x = 5 可化为 x = −5。
6. Equations with Fractions | 含分数的方程
Fractions can feel tricky, but clearing them early makes equations much simpler. Multiply every term on both sides by the lowest common multiple (LCM) of all denominators.
分数可能会让人觉得棘手，但尽早消去分母可以让方程变得简单许多。将两边每一项都乘以所有分母的最小公倍数 (LCM)。
Solve x/4 + 1 = 3. Multiply everything by 4: 4 × (x/4) + 4 × 1 = 4 × 3, which gives x + 4 = 12, so x = 8.
解方程 x/4 + 1 = 3。每一项都乘以 4：4 × (x/4) + 4 × 1 = 4 × 3，得到 x + 4 = 12，因此 x = 8。
For something like (x + 2)/3 = 5, you can either multiply both sides by 3 immediately, or think of it as "x plus 2 all divided by 3 equals 5". Multiplying by 3 gives x + 2 = 15, and then x = 13.
对于像 (x + 2)/3 = 5 这样的方程，你可以立刻将两边乘以 3，或者把它理解为“x 加 2 的和除以 3 等于 5”。乘以 3 得到 x + 2 = 15，进而 x = 13。
Technique: For x/2 + x/3 = 5, multiply by LCM of 2 and 3 (which is 6) to get 3x + 2x = 30.
LCM 技巧：对于 x/2 + x/3 = 5，乘以 2 和 3 的最小公倍数 6，得到 3x + 2x = 30。
7. Checking Your Solution | 检验你的解
Always substitute your answer back into the original equation to verify it works. This catches arithmetic mistakes and builds confidence.
始终将你的答案代回原方程，验证它是否成立。这能发现计算错误，并树立信心。
If you solved 2x − 4 = 10 and got x = 7 14 − 4 = 10. It matches, so the answer is correct. Get into this habit for every problem.
如果你解出了 2x − 4 = 10 并得到 x = 7，检验一下：2(7) − 4 = 14 − 4 = 10。结果吻合，所以答案正确。要养成每个问题都检验的习惯。
If the check fails, retrace your steps. Often a small sign error or an incorrectly expanded bracket is the culprit. Checking is the best debugging tool you have.
如果检验不通过，重新回溯你的步骤。通常一个小小的符号错误或者括号展开错误是罪魁祸首。检验是你拥有的最好的调试工具。
8. Word Problems and Forming Equations | 应用题与建立方程
Many real-life situations can be modelled with linear equations. Start by reading the problem carefully and assigning a variable to the unknown quantity.
许多现实情境都可以用线性方程来建模。先仔细阅读问题，然后为未知量设一个变量。
For example: "Three more than twice a number is 19. Find the number." Let the number be n. Then 2n + 3 = 19. Solve to get n = 8.
例如：“一个数的两倍加三等于 19。求这个数。” 设这个数为 n。那么 2n + 3 = 19。解方程得到 n = 8。
Always define your variable clearly. If the question asks for two things, express one in terms of the other. Practice translating words like "sum", "product", "less than", and "more than" into mathematical symbols.
始终清晰地定义你的变量。如果问题要求两个量，把其中一个用另一个表示出来。多练习将诸如“和”、“积”、“少于”、“多于”等词语转化为数学符号。
9. Common Mistakes and How to Avoid Them | 常见错误及其避免方法
Mistake 1: Forgetting to do the same operation on both sides. For example, subtract 5 from the left but not from the right. Use the balancing method literally – imagine a scale.
错误 1：忘记在等号两边执行相同的操作。例如，左边减了 5，但右边没减。要如实使用天平法——在脑海中想象一架天平。
Mistake 2: Incorrect sign when expanding brackets with a negative multiplier. Always write down the expanded form step by step.
错误 2：带有负乘数的括号展开时符号出错。始终一步一步地写出展开后的形式。
Mistake 3: Dividing or multiplying incorrectly when dealing with fractions. Check if you need to multiply every term by the denominator.
错误 3：在处理分数时进行错误的乘除运算。检查是否需要把每一项都乘以分母。
Mistake 4: Losing track of the variable when moving terms. Write each new equivalent equation on a fresh line so you can see the transformation clearly.
错误 4：移项时丢失了变量。把每个新的等价方程写在单独一行，这样你可以清晰地看到变换过程。
| Common Error | How to Fix |
| 3(x − 2) → 3x − 2 | Multiply both terms: 3x − 6 |
| Subtracting 5 from one side only | Do the same to both sides |
| 常见错误 | 纠正方法 |
| 3(x − 2) → 3x − 2 | 两项都乘：3x − 6 |
| 只从一边减去 5 | 等号两边做相同操作 |
10. Steps Summary and Practice Ideas | 步骤总结与练习建议
To solve any linear equation, follow a clear sequence: clear fractions if any, expand brackets, collect like terms on each side, move variable terms to one side and constants to the other, combine, then isolate the variable.
要解任何一个线性方程，请遵循清晰的步骤序列：如果有分数就先消去分母，展开括号，分别在两边合并同类项，将变量项移到一边、常数项移到另一边，合并，然后解出变量。
Practise with a mix of difficulty levels: one-step equations, two-step equations, equations with brackets, and equations with unknowns on both sides. Start each problem by writing down the original equation and finish by checking your answer.
练习时要混合不同难度级别：一步方程、两步方程、带括号的方程以及未知数在两边的方程。每个问题开始时写下原方程，最后检验答案。
Gradually increase complexity. Try equations like 2(3x − 1) = 4(x + 5) − 2x. Expand, simplify, and solve. With persistent practice, you will recognise patterns and solve confidently and quickly.
逐步增加复杂度。尝试诸如 2(3x − 1) = 4(x + 5) − 2x 的方程。先展开、化简，再求解。通过持续练习，你将能识别模式，自信而快速地解题。
Published by TutorHao | 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