📚 Solving Linear Equations for KS3 Cambridge | KS3剑桥数学：解一元一次方程
Linear equations are one of the most important skills in the Cambridge KS3 Mathematics curriculum. They appear in algebra, geometry, and word problems across Year 7 to Year 9. This article explains how to solve them step by step, check answers, and avoid common errors.
一元一次方程是剑桥 KS3 数学课程中最重要的技能之一。它们出现在代数、几何以及 Year 7 到 Year 9 的文字题中。本文将逐步讲解如何解方程、检验答案并避免常见错误。
1. What is a Linear Equation? | 什么是一元一次方程？
A linear equation is an equation where the unknown variable, often written as x, is only raised to the power of 1. For example, 2x + 3 = 11 is linear. If you see x² or x³, it is not a linear equation at KS3 level.
一元一次方程是指未知数（通常用 x 表示）的指数为 1 的方程。例如 2x + 3 = 11 就是一元一次方程。如果出现 x² 或 x³，就不是 KS3 阶段的一元一次方程。
Linear equations can look different, but they all mean the same thing: two expressions have the same value. Your job is to find the value of the unknown that makes the equation true.
一元一次方程可以有不同形式，但它们都表示同一个意思：两个表达式的值相等。你的任务是找出使方程成立的未知数的值。
2. The Balancing Method | 天平法（等式平衡）
Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side. This keeps the equation true. If you add 5 to the left, add 5 to the right.
把方程想象成一台天平。你对一边做的任何运算，必须对另一边也做同样的运算，这样等式才能保持成立。如果左边加 5，右边也要加 5。
If a = b, then a + c = b + c, a – c = b – c, a × c = b × c, a ÷ c = b ÷ c (c ≠ 0)
This rule is the basis of every method shown below. Never add, subtract, multiply, or divide only one side of an equation.
这条规则是下面所有方法的基础。绝不能只对等式的一边进行加、减、乘或除。
3. Solving One-Step Equations | 解一步方程
Use inverse operations to isolate the unknown. For x + 4 = 9, subtract 4 from both sides to get x = 5. For 3x = 18, divide both sides by 3 to get x = 6.
使用逆运算将未知数单独留在一边。对于 x + 4 = 9，两边同时减 4，得 x = 5。对于 3x = 18，两边同时除以 3，得 x = 6。
If the equation is x ÷ 4 = 5, multiply both sides by 4 to get x = 20. The inverse of addition is subtraction, and the inverse of multiplication is division.
如果方程是 x ÷ 4 = 5，两边同时乘以 4，得 x = 20。加法的逆运算是减法，乘法的逆运算是除法。
- x + 7 = 15 → x = 8
- x – 3 = 9 → x = 12
- 5x = 35 → x = 7
- x ÷ 6 = 4 → x = 24
4. Solving Two-Step Equations | 解两步方程
For an equation such as 2x + 3 = 11, undo the addition or subtraction first, then undo the multiplication or division. Subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4.
对于 2x + 3 = 11 这样的方程，先处理加减法，再处理乘除法。两边先减 3，得 2x = 8，然后两边除以 2，得 x = 4。
2x + 3 = 11 → 2x = 8 → x = 4
This order is important because the term 2x is being grouped as one value before 3 is added. You must remove the 3 first to reveal 2x.
这个顺序很重要，因为 2x 在被加上 3 之前是作为一个整体存在的。你必须先去掉 3，才能得到 2x。
5. Equations with Brackets | 含括号的方程
Always expand brackets first unless the bracket is being divided or multiplied by a single number. For 3(x + 2) = 21, expand to 3x + 6 = 21, then subtract 6 and divide by 3 to get x = 5.
通常先展开括号，除非括号被一个数除或乘。对于 3(x + 2) = 21，展开为 3x + 6 = 21，然后两边减 6，再除以 3，得 x = 5。
3(x + 2) = 21 → 3x + 6 = 21 → 3x = 15 → x = 5
If you see a negative outside the bracket, be careful with signs. For -2(x – 4) = 10, expand to -2x + 8 = 10, then solve to find x = -1.
如果括号外是负数，要注意符号。对于 -2(x – 4) = 10，展开为 -2x + 8 = 10，然后解方程得 x = -1。
6. Variables on Both Sides | 两边含未知数的方程
When the unknown appears on both sides, collect the variable terms on one side first. Solve 5x – 7 = 2x + 5 by subtracting 2x from both sides to get 3x – 7 = 5. Then add 7 and divide by 3 to find x = 4.
当未知数出现在等式两边时，先把含未知数的项合并到同一边。解 5x – 7 = 2x + 5，两边先减 2x，得 3x – 7 = 5，然后加 7，再除以 3，得 x = 4。
5x – 7 = 2x + 5 → 3x – 7 = 5 → 3x = 12 → x = 4
A common choice is to subtract the smaller variable term from both sides. This keeps the coefficient of x positive, which is easier to work with.
一个常见做法是从两边减去较小的未知数项。这样可以使 x 的系数保持为正，计算起来更容易。
7. Equations with Fractions | 含分数的方程
To clear fractions, multiply every term on both sides by the denominator. For x/3 + 2 = 6, multiply all terms by 3 first: x + 6 = 18, then subtract 6 to get x = 12.
为了消除分数，将两边每一项都乘以分母。对于 x/3 + 2 = 6，先把所有项都乘以 3：x + 6 = 18，然后两边减 6，得 x = 12。
x/3 + 2 = 6 → x + 6 = 18 → x = 12
If there is more than one fraction, multiply by the lowest common denominator of all fractions. For x/2 + x/3 = 5, multiply by 6: 3x + 2x = 30, so 5x = 30 and x = 6.
如果有多个分数，要乘以所有分母的最小公分母。对于 x/2 + x/3 = 5，两边乘以 6：3x + 2x = 30，所以 5x = 30，x = 6。
8. Forming Equations from Word Problems | 从文字题建立方程
Translate words into algebra. The sentence “I think of a number, double it and add 7; the answer is 19” becomes 2n + 7 = 19. Define the variable clearly before writing the equation.
把文字翻译成代数。句子“我想一个数，把它乘 2 再加 7，结果是 19”可以写成 2n + 7 = 19。在写方程前要先清楚地定义变量。
2n + 7 = 19 → 2n = 12 → n = 6
Geometry also uses linear equations. In a triangle, if the angles are x°, 2x°, and 3x°, the sum is 180°, so x + 2x + 3x = 180. This gives 6x = 180 and x = 30.
几何中也会用到一元一次方程。在一个三角形中，如果三个角分别是 x°、2x° 和 3x°，则内角和为 180°，所以 x + 2x + 3x = 180。由此得 6x = 180，x = 30。
9. Checking Your Answer | 检验答案
Always substitute your value back into the original equation. If 2x + 3 = 11 and x = 4, then 2(4) + 3 = 8 + 3 = 11. Since both sides match, the answer is correct.
始终把你求得的值代回原方程进行检验。如果 2x + 3 = 11 且 x = 4，那么 2(4) + 3 = 8 + 3 = 11。两边相等，答案就是正确的。
| Original equation | Solution | Check |
| 4x – 5 = 19 | x = 6 | 4(6) – 5 = 24 – 5 = 19 |
| 2(x + 3) = 16 | x = 5 | 2(5 + 3) = 2(8) = 16 |
The check step is especially useful in exams because it catches sign errors and arithmetic mistakes before you move on.
检验这一步在考试中特别有用，因为它能在你继续做下一题前发现符号错误和计算错误。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
The first common mistake is operating on only one side of the equation. Remember that any operation must be applied to both sides to keep the balance.
第一个常见错误是只对等式的一边进行运算。记住，任何运算都必须同时作用于两边，才能保持等式平衡。
The second mistake is sign errors when moving terms. For example, in 3x – 4 = 2x + 6, subtracting 2x from both sides gives x – 4 = 6, not x + 4 = 6.
第二个常见错误是移项时出现符号错误。例如在 3x – 4 = 2x + 6 中，两边减 2x 得到 x – 4 = 6，而不是 x + 4 = 6。
Finally, remember to check whether the unknown is just x, not x². Linear equations only include x to the
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