📚 Solving Linear Equations | 解一元一次方程
Linear equations are the building blocks of algebra. In KS3 mathematics, understanding how to solve them gives you a powerful tool for finding unknown values and modelling real-world problems. This article walks through the key concepts, from simple one-step equations to more complex ones with brackets and variables on both sides.
线性方程是代数的基础。在 KS3 数学中，掌握如何解线性方程，你就有了一个强大的工具——来寻找未知数的值，并建立现实世界问题的模型。本文将从简单的一元一次方程讲起，逐步覆盖带括号和变量在等式两侧的更复杂方程，梳理其中的核心概念。
1. What is a Linear Equation? | 什么是线性方程？
ation, while x² + 3 = 12 is not. The graph of a linear equation is always a straight line.
线性方程是指未知数（通常用 x 或 y 等字母表示）的指数仅为 1 的方程。方程中不出现变量的平方、立方或根号。例如，2x + 5 = 15 是线性方程，而 x² + 3 = 12 则不是。线性方程的图像总是一条直线。
At KS3 level, you mainly solve equations in one variable, looking for the value that makes the equation balance. The solution, or root, is the number you substitute into the equation to make both sides equal.
在 KS3 阶段，你主要需要解一元方程，即寻找能让等式成立的未知数的值。这个值——也称为解或根——代入方程后，等式两边会相等。
2. The Balance Method | 平衡法
Think of an equation as a set of scales. Whatever you do to one side of the equation, you must do exactly the same to the other side to keep the scales balanced. This is the golden rule of solving equations. If you add 3 to the left-hand side, you must also add 3 to the right-hand side.
你可以把等式想象成一个天平。对等式的一边做了什么运算，对另一边也必须做完全相同的运算，才能保持天平平衡。这是解方程的黄金法则。如果在左边加上 3，那么右边也必须加上 3。
− 4, which simplifies to x = 5. The balance method helps you undo operations step by step.
例如，要解 x + 4 = 9，你需要在两边同时减去 4，因为你要把 x 独立出来。这样得到 x + 4 − 4 = 9 − 4，化简后 x = 5。平衡法可以帮你一步步逆向消去运算。
3. Solving One-Step Equations | 解一步方程
→ divide both sides by 4 → a = 5. Division: b ÷ 6 = 3 → multiply both sides by 6 → b = 18.
一步方程只涉及一次运算就可以把变量分离出来。加法形式：x + 7 = 12 → 两边同时减去 7 → x = 5。减法形式：y − 3 = 10 → 两边同时加 3 → y = 13。乘法形式：4a = 20 → 两边同时除以 4 → a = 5。除法形式：b ÷ 6 = 3 → 两边同时乘以 6 → b = 18。
It is important to use the inverse operation: subtraction undoes addition, addition undoes subtraction, division undoes multiplication, and multiplication undoes division. Always write the operation you are performing on both sides neatly to avoid mistakes.
关键是运用逆运算：减法消去加法，加法消去减法，除法消去乘法，乘法消去除法。要整洁地写出在两边同时进行的运算，从而避免失误。
4. Solving Two-Step Equations | 解两步方程
ys reverse the order of operations: take care of addition or subtraction before tackling multiplication or division.
在两步方程中，例如 2x + 3 = 11，你需要逆序消去两种运算。先在两边减去 3：2x = 8；然后在两边除以 2：x = 4。务必逆着运算顺序进行：先处理加减，再处理乘除。
−4z + 5 = 25, you subtract 5: −4z = 20, then divide by −4: z = −5. Be careful with signs.
再如 3y − 7 = 14。两边加 7：3y = 21；然后除以 3：y = 7。如果变量项的系数是负数，例如 −4z + 5 = 25，先减去 5：−4z = 20，再除以 −4：z = −5。要格外注意符号。
5. Equations with Brackets | 带括号的方程
3x + 6 = 21. Then solve as a two-step equation: subtract 6 → 3x = 15; divide by 3 → x = 5.
如果方程含有括号，例如 3(x + 2) = 21，主要有两种解法。第一种是展开括号：3 × x + 3 × 2 = 3x + 6，方程变为 3x + 6 = 21。然后按两步方程求解：减 6 → 3x = 15；除以 3 → x = 5。
÷ 3 → x + 2 = 7 → x = 5. Both methods give the same answer; expanding is safer in general.
第二种方法是先两边同时除以括号外的系数，但这仅在右边可被整除时才简便。此例中：3(x + 2) ÷ 3 = 21 ÷ 3 → x + 2 = 7 → x = 5。两种方法结果一致；一般情况下，展开括号更稳妥。
6. Equations with Unknowns on Both Sides | 未知数在两侧的方程
er. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8. Divide by 2: x = 4.
当变量出现在等式两边时，例如 5x + 2 = 3x + 10，目标是把所有含变量的项移到一边，常数项移到另一边。两边同时减去 3x：2x + 2 = 10；再减 2：2x = 8；除以 2：x = 4。
vide by 3: y = −5. Check your solution by substituting it back into the original equation.
一般先消去较小的变量项，使系数保持正号。对于 2y − 7 = 5y + 8，两边减去 2y：−7 = 3y + 8；再减 8：−15 = 3y；除以 3：y = −5。将解代回原方程可以检验是否正确。
7. Dealing with Fractions in Equations | 处理含分数的方程
everything by 3: x + 6 = 15; then subtract 6: x = 9. This clears the fraction immediately.
方程中也可能含有分数，例如 (x/3) + 2 = 5。一种高效的方法是两边每一项同时乘以最小公分母。这里，所有项乘以 3：x + 6 = 15；再减 6：x = 9。这样可以立刻消去分母。
ltiple fractions, find the LCM of all denominators. Always apply the multiplication to every term, including whole numbers.
若方程为 (2x + 1)/4 = 3，两边乘 4：2x + 1 = 12；接着求解：2x = 11 → x = 5.5。当有多个分数时，求出所有分母的最小公倍数。切记要把乘法应用于每一项，包括整数项。
8. Checking Your Solution | 检验解
= 20. They are not equal; therefore, a mistake was made. Re-solving gives x = 12/5 = 2.4.
每次求出解之后，都应该将它代回原方程，验证是否成立。例如，你解 4(2x − 1) = 3x + 8 得出 x = 4，将 x = 4 代入两边。左边：4(2×4 − 1) = 4(8 − 1) = 28；右边：3×4 + 8 = 12 + 8 = 20。两边不相等，说明有误。重新计算可得 x = 12/5 = 2.4。
Checking prevents losing marks on simple arithmetic errors. If the substituted value makes both sides equal, you can be confident your solution is correct. This habit is especially important during exams.
检验可以避免因简单计算错误而丢分。如果代值后两边相等，你就能确信解是正确的。养成这个习惯对考试尤为重要。
9. Common Mistakes to Avoid | 应避免的常见错误
Forgetting to apply an operation to all terms. . When you divide, divide every term on both sides. For 3x + 6 = 9, dividing only the 3x by 3 but not the 6 gives x + 6 = 3, which is wrong.
忘记对每一项进行运算。 在相除时，要对两边每一项都除。对于 3x + 6 = 9，如果只把 3x 除以 3 而不管 6，就会得到错误的 x + 6 = 3。Incorrect distribution when expanding brackets. Remember to multiply the outside number with every term inside the bracket, including negative signs. 2(3 − x) expands to 6 − 2x, not 6 − x.
展开括号时分配律错误。 要记得把外面的数与括号内的每一项相乘，符号也要带上。2(3 − x) 展开后是 6 − 2x，而不是 6 − x。Sign errors when moving terms. Changing the side of a term changes its sign. If you move +5 to the other side, it becomes −5.
移项时符号错误。 将一项移到等号另一边要变号。+5 移项后变为 −5。
10. Word Problems Leading to Linear Equations | 线性方程文字应用题
Many real-world situations can be modelled by linear equations. For example: "The sum of three consecutive numbers is 72. Find the numbers." mplify: 3n + 3 = 72. Subtract 3: 3n = 69. Divide by 3: n = 23. The numbers are 23, 24, 25.
许多现实情境都可以用线性方程建模。例如：“三个连续整数的和是 72，求这三个数。”设第一个数为 n，则 n + (n + 1) + (n + 2) = 72。化简：3n + 3 = 72；减 3：3n = 69；除以 3：n = 23。这三个数是 23、24、25。
Another common type: " s old as his daughter. In 10 years, the sum of their ages will be 68. How old is David now?" Let daughter's age = d, then David' In 10 years: (2d + 10) + (d + 10) = 68 → 3d + 20 = 68 → 3d = 48 → d = 16, so David is 32.
另一常见题型：“戴维的年龄是他女儿的两倍。10 年后，两人的年龄之和为 68。戴维现在多少岁？”设女儿年龄为 d，戴维为 2d。10 年后：(2d + 10) + (d + 10) = 68 → 3d + 20 = 68 → 3d = 48 → d = 16，所以戴维 32 岁。
Always define your variable clearly and translate the written scenario into an algebraic equation step by step. Then use your solving skills to find the unknown.
务必清晰地定义变量，将文字情境一步步转化为代数方程。然后用你学到的解方程技巧找到未知数。
11. More Practice: Mixed Examples | 更多练习：混合示例
Try solving these equations to test your understanding:
试着解以下方程，检验你的理解程度：
| Equation | Steps | Solution |
| 5(x − 3) = 2x + 9 | Expand: 5x − 15 = 2x + 9 → subtract 2x: 3x − 15 = 9 → add 15: 3x = 24 → divide 3: x = 8 | x = 8 |
| (2y)/3 + 4 = 10 | Multiply by 3: 2y + 12 = 30 → subtract 12: 2y = 18 → divide 2: y = 9 | y = 9 |
| 7 − 2x = 3x − 8 | Add 2x: 7 = 5x − 8 → add 8: 15 = 5x → divide 5: x = 3 | x = 3 |
Work through each example on your own before comparing with the solution. Pay close attention to the order of operations and the signs as you rearrange terms.
在对照答案之前，先自己尝试完成每一道题。移项时要特别注意运算顺序和符号。
12. Key Takeaways and Further Steps | 关键要点与后续学习
Mastering linear equations at KS3 provides a strong foundation for solving inequalities, simultaneous equations, and quadratic equations later on. The core principles—doing the same to both sides, collecting like terms, and checking solutions—remain relevant throughout your mathematical journey.
在 KS3 阶段掌握线性方程，可以为后续学习不等式、联立方程和二次方程打下坚实的基础。两边同运算、合并同类项、检验解等核心原则，在你整个数学学习过程中都会一直用到。
Remember: every equation is a puzzle, and every step brings you closer to the solution. Practice regularly, learn from your mistakes, and you will soon solve even the trickiest linear equations with confidence.
请记住：每个方程都是一道谜题，每一步都让你离答案更近。坚持练习，从错误中学习，你很快就能自信地解决最难的一元一次方程。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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