📚 Linear Equations and Inequalities | 线性方程与不等式
Welcome to this KS3 Cambridge Mathematics revision guide. In this article, we will explore how to solve linear equations step by step, how to handle brackets and fractions, and how to represent inequalities on a number line. These skills form the backbone of algebra at KS3 and prepare you for IGCSE and beyond.
欢迎阅读本 KS3 剑桥数学复习指南。在本文中，我们将逐步学习如何解一元一次方程、如何处理括号与分数，以及如何在数轴上表示不等式。这些技能是 KS3 代数的基础，能为你未来学习 IGCSE 及更高年级内容做好准备。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is an algebraic statement in which two expressions are equal, and the unknown variable has an exponent of 1. For example, 3x + 5 = 14 is a linear equation because the variable x is not squared or cubed.
线性方程是一个代数等式，表示两个表达式相等，并且未知变量的指数为 1。例如，3x + 5 = 14 就是一个线性方程，因为变量 x 没有平方或立方。
In a linear equation, the solution is the value of the variable that makes the equation true. You can check a solution by substituting it back into the original equation.
在线性方程中，解就是使等式成立的变量值。你可以将解代回原方程来检验是否正确。
- General form: ax + b = c, where a, b, and c are constants.
- 一般形式：ax + b = c，其中 a、b、c 是常数。
- If a term contains x² or x³, it is not linear.
- 如果某一项含有 x² 或 x³，它就不是线性方程。
- Linear means the graph of the equation is a straight line.
- 线性意味着方程的图像是一条直线。
3x + 5 = 14
x = 3 → check: 3(3) + 5 = 14
2. Balancing Method | 天平法（等式平衡法）
Think of an equation as 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 solving equations.
把方程想象成一个天平。无论你对一边做什么操作，都必须对另一边做同样的操作，才能保持平衡。这是解方程的黄金法则。
To isolate the variable, you can add, subtract, multiply, or divide both sides by the same non-zero number. This keeps the equation equivalent and does not change the solution.
为了分离变量，你可以在等式两边加上、减去、乘以或除以同一个非零数。这会使方程保持等价，并且不会改变解。
For example, to solve x + 7 = 15, subtract 7 from both sides.
例如，解 x + 7 = 15，两边同时减去 7。
x + 7 − 7 = 15 − 7
x = 8
Similarly, to solve x − 9 = 3, add 9 to both sides to get x = 12. Every operation must be applied to the whole side, not just one term.
类似地，解 x − 9 = 3 时，两边同时加上 9，得到 x = 12。每一步操作都必须作用于整个一边，而不仅仅是某一项。
3. Solving Two-Step Equations | 解两步方程
Many KS3 equations require two operations. The key is to undo the addition or subtraction first, then undo the multiplication or division. This reverse order follows the opposite of the order of operations.
许多 KS3 方程需要两步运算。关键是先消去加法或减法，再消去乘法或除法。这个逆序与运算顺序相反。
Example: solve 4x − 3 = 17.
例题：解 4x − 3 = 17。
- Step 1: Add 3 to both sides to get 4x = 20.
- 步骤 1：两边同时加 3，得到 4x = 20。
- Step 2: Divide both sides by 4 to get x = 5.
- 步骤 2：两边同时除以 4，得到 x = 5。
4x − 3 + 3 = 17 + 3
4x = 20 → x = 5
Always verify your answer by substituting x = 5 back into the original equation: 4(5) − 3 = 20 − 3 = 17.
一定要将 x = 5 代回原方程进行检验：4(5) − 3 = 20 − 3 = 17。
Another example is 7 + 2x = 19. Subtract 7 first to get 2x = 12, then divide by 2 to find x = 6.
另一个例子是 7 + 2x = 19。先减去 7，得到 2x = 12，再除以 2，得到 x = 6。
4. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive law. Multiply each term inside the bracket by the term outside.
当方程中含有括号时，先用分配律展开。将括号外的项乘以括号内的每一项。
Example: solve 3(x + 4) = 27.
例题：解 3(x + 4) = 27。
3x + 12 = 27
Then subtract 12 from both sides and divide by 3 to find x = 5.
然后两边同时减去 12，再除以 3，得到 x = 5。
If there is a negative outside the bracket, be careful with signs: −2(x − 5) becomes −2x + 10.
如果括号外是负数，要注意符号：−2(x − 5) 展开后为 −2x + 10。
−2(x − 5) = −2x + 10
After expanding, solve the equation using the balancing method. Never divide by the number outside the bracket before expanding, because the bracket must be treated as a single group first.
展开后，用天平法解方程。切勿在展开前先除以括号外的数，因为括号必须首先作为一个整体来处理。
5. Equations with Variables on Both Sides | 变量在等式两边的方程
Some equations have the unknown on both sides, such as 5x + 2 = 2x + 14. Start by collecting like terms so the variable appears on only one side.
有些方程两边都含有未知数，例如 5x + 2 = 2x + 14。首先合并同类项，使变量只出现在一边。
Subtract 2x from both sides to get 3x + 2 = 14. Then subtract 2 and divide by 3, giving x = 4.
两边同时减去 2x，得到 3x + 2 = 14。再减去 2，然后除以 3，得到 x = 4。
5x + 2 = 2x + 14
5x − 2x + 2 = 2x − 2x + 14
3x + 2 = 14 → x = 4
It is often easier to move the smaller variable term to the larger side, but either side works as long as you balance every step.
通常把较小的变量项移到较大的一边更容易，但只要每一步都保持平衡，移到哪一边都可以。
If the variable term becomes negative, for example −x = −6, multiply both sides by −1 to get x = 6. The same rule applies: an equation stays true when both sides are multiplied by the same value.
如果变量项变成负数，例如 −x = −6，两边同时乘以 −1，得到 x = 6。同样的规则适用：两边乘以同一个数时，等式仍然成立。
6. Equations with Fractions | 含分数的方程
To solve an equation with fractions, multiply every term by the lowest common denominator (LCD) to clear the fractions. Then solve the resulting linear equation.
解含分数的方程时，将每一项乘以最小公分母（LCD），先消去分母，然后解得到的线性方程。
Example: solve x/3 + 2 = 5.
例题：解 x/3 + 2 = 5。
Multiply all terms by 3: x + 6 = 15, so x = 9.
所有项乘以 3：x + 6 = 15，所以 x = 9。
For equations like (x + 1)/4 = 3, multiply both sides by 4 to get x + 1 = 12, then x = 11.
对于 (x + 1)/4 = 3 这类方程，两边乘以 4，得到 x + 1 = 12，然后 x = 11。
x/3 + 2 = 5 → x + 6 = 15 → x = 9
If there are two different denominators, such as x/2 + x/3 = 5, multiply every term by 6. This gives 3x + 2x = 30, so 5x = 30 and x = 6.
如果有两个不同的分母，例如 x/2 + x/3 = 5，将每一项乘以 6。得到 3x + 2x = 30，所以 5x = 30，x = 6。
7. Introduction to Inequalities | 不等式入门
An inequality compares two expressions using symbols: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Unlike an equation, an inequality usually has many solutions.
不等式用符号比较两个表达式：>（大于）、<（小于）、≥（大于或等于）、≤（小于或等于）。与方程不同，不等式通常有很多个解。
For example, x > 4 means x can be any number larger than 4, such as 4.1, 5, or 100, but not 4 itself.
例如，x > 4 表示 x 可以是任何大于 4 的数，如 4.1、5 或 100，但不能是 4 本身。
- x < 3: values less than 3, excluding 3.
- x < 3：小于 3 的值，不包括 3。
- x ≥ −2: values greater than or equal to −
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