📚 Solving Linear Equations for Cambridge KS3 Mathematics | 剑桥KS3数学：解线性方程
s how to solve one-step, two-step, and more challenging linear equations using the balancing method, and how to check your answer correctly.
线性方程是剑桥KS3数学中最重要的基础内容之一。从简单的数字谜题到现实生活中的应用题，它几乎出现在每一个代数主题中。本文基于 p321_1.pdf 所涵盖的技能，讲解如何使用平衡法解一步方程、两步方程以及更复杂的线性方程，并正确地检验答案。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is a mathematical statement that shows two expressions are equal, and the unknown variable has a power of 1. In KS3, the unknown is usually written as x, y, or a letter like a.
线性方程是一种数学语句，它表示两个表达式相等，且未知数的最高次数为 1。在KS3阶段，未知数通常用 x、y 或 a 等字母表示。
- x + 3 = 7 — 'x' has power 1, so it is linear.
- 2x − 5 = 9 — the coefficient 2 does not change the power of x.
- x² + 1 = 5 — this is not linear because the power of x is 2.
x + 3 = 7 → x = 7 − 3 → x = 4
x + 3 = 7 → x = 7 − 3 → x = 4
2. The Balancing Method | 平衡法
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 and allows you to isolate the unknown.
把方程想象成一个天平：你对一边做什么，就必须对另一边做同样的操作。这样方程才能保持成立，并且帮助你分离出未知数。
If we have x + 5 = 12, we can subtract 5 from both sides to keep the balance.
如果我们有 x + 5 = 12，我们可以两边同时减去 5 来保持平衡。
x + 5 − 5 = 12 − 5 → x = 7
x + 5 − 5 = 12 − 5 → x = 7
3. Solving One-Step Equations | 解一步方程
One-step equations need just one inverse operation to solve. The inverse of addition is subtraction, and the inverse of multiplication is division.
一步方程只需要一步逆运算即可求解。加法的逆运算是减法，乘法的逆运算是除法。
- x + 4 = 10 → subtract 4 → x = 6
- x − 3 = 8 → add 3 → x = 11
- 5x = 20 → divide by 5 → x = 4
- x ÷ 6 = 2 → multiply by 6 → x = 12
The table below summarises how to undo each operation.
下表总结了如何撤销每种运算。
| Operation | Inverse Operation |
|---|---|
| + | − |
| − | + |
| × | ÷ |
| ÷ | × |
4. Solving Two-Step Equations | 解两步方程
Two-step equations involve two operations, such as multiplication and addition. Always undo the addition or subtraction first, then undo the multiplication or division.
两步方程涉及两种运算，例如乘法和加法。一定要先撤销加法或减法，再撤销乘法或除法。
Solve 2x + 3 = 11. First subtract 3 from both sides, then divide both sides by 2.
解方程 2x + 3 = 11。首先两边同时减去 3，然后两边同时除以 2。
2x + 3 = 11 → 2x = 8 → x = 4
2x + 3 = 11 → 2x = 8 → x = 4
This reverse order follows the inverse of the order of operations: undo brackets, then division/multiplication, then addition/subtraction.
这种逆序对应运算顺序的逆过程：先处理括号，再处理除法或乘法，最后处理加法或减法。
5. Expanding Brackets First | 先展开括号
When an equation contains brackets, expand them first using the distributive law. This turns the equation into a familiar two-step form.
当方程含有括号时，首先使用分配律展开括号。这样方程就变成了熟悉的两步形式。
For 3(x + 2) = 15, expand to get 3x + 6 = 15, then subtract 6 and divide by 3.
对于 3(x + 2) = 15，先展开得到 3x + 6 = 15，然后减去 6，再除以 3。
3(x + 2) = 15 → 3x + 6 = 15 → 3x = 9 → x = 3
3(x + 2) = 15 → 3x + 6 = 15 → 3x = 9 → x = 3
6. Equations with Unknowns on Both Sides | 两边有未知数
If the unknown appears on both sides, collect the x terms on one side and the number terms on the other. You can do this by adding or subtracting the same x term from both sides.
如果未知数出现在等号两边，就把含 x 的项移到一边，把数字项移到另一边。你可以通过在两边同时加上或减去同一个含 x 的项来实现。
e 5x + 2 = 2x + 11. Subtract 2x from both sides, then subtract 2, and finally divide by 3.
解方程 5x + 2 = 2x + 11。两边同时减去 2x，再减去 2，最后除以 3。
5x + 2 = 2x + 11 → 3x + 2 = 11 → 3x = 9 → x = 3
5x + 2 = 2x + 11 → 3x + 2 = 11 → 3x = 9 → x = 3
7. Equations Involving Fractions | 涉及分数的方程
When an equation contains a fraction, multiply both sides by the denominator first. This removes the fraction and gives a simpler equation.
当方程含有分数时，首先两边同时乘以分母。这样可以去掉分数，得到更简单的方程。
Solve x/4 + 1 = 3. First subtract 1 from both sides, then multiply both sides by 4.
解方程 x/4 + 1 = 3。首先两边同时减去 1，然后两边同时乘以 4。
x/4 + 1 = 3 → x/4 = 2 → x = 8
x/4 + 1 = 3 → x/4 = 2 → x = 8
ator such as 2x/5 = 6, multiply by 5 first to get 2x = 30, then divide by 2 to get x = 15.
如果分数的分子是 2x/5 = 6，先乘以 5 得到 2x = 30，再除以 2 得到 x = 15。
8. Checking Your Answer | 检验答案
After solving an equation, always substitute your answer back into the original equation. If the left side equals the right side, your answer is correct.
解完方程后，一定要把答案代回原方程。如果左边等于右边，说明答案是正确的。
in 2x + 3 = 11, the left side becomes 2(4) + 3 = 8 + 3 = 11, which matches the right side.
对于 2x + 3 = 11 中的 x = 4，左边为 2(4) + 3 = 8 + 3 = 11，与右边相等。
This checking step is especially important in exam questions, where a small arithmetic mistake can cost marks.
在考试题中，检验这一步尤其重要，因为一个小小的计算错误就可能导致失分。
9. Common Mistakes to Avoid | 常见错误
Many KS3 students lose marks on linear equations because of avoidable errors. Be careful with signs, especially when subtracting a negative number.
许多KS3学生在解线性方程时因为一些可以避免的错误而失分。注意符号，尤其是减去负数的情况。
- Forgetting to do the same operation on both sides.
- Forgetting to divide by the coefficient, so 2x = 8 becomes x = 8 instead of x = 4.
- Sign errors, such as writing x − 5 = 2 so x = 2 − 5 = −3 instead of x = 7.
- Not expanding brackets correctly: 3(x + 2) is not 3x + 2.
Always write each step clearly on a new line. This makes it easier to spot mistakes and to revise your method.
每一步都要清晰地写在新的一行。这样更容易发现错误，也方便检查你的方法。
10. Exam-Style Practice and Tips | 考试题型与技巧
Exam questions often ask you to solve an equation, then use the value in a word problem. Read the question twice and highlight key information such as 'total', 'difference', or 'equals'.
考试题常常要求解出一个方程，然后在应用题中使用这个值。读题两遍，标出关键信息，例如 'total'、'difference' 或 'equals'。
Try this Cambridge KS3 style problem: 'I think of a number. I multiply it by 3, then add 7. The result is 28. Find the number.'
试试这道剑桥KS3风格的题目：“我想了一个数。把它乘以3，再加7。结果是28。求这个数。”
3x + 7 = 28 → 3x = 21 → x = 7
3x + 7 = 28 → 3x = 21 → x = 7
For top marks, show all working, write the answer clearly, and check it in the original equation. These habits will help you in KS3 and beyond.
想要拿高分，就要写出完整的步骤，清晰地写出答案，并在原方程中检验。这些习惯会在KS3及以后的学习中帮助你。
Published by TutorHao | Maths 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