📚 Solving Linear Equations and Inequalities | 解一元一次方程与不等式
Linear equations are one of the most important building blocks in KS3 mathematics. They allow you to find an unknown value by using inverse operations in a logical order. In this article, we will explore how to solve one-step, two-step, and multi-step linear equations, and then extend these ideas to linear inequalities.
线性方程是 KS3 数学中最重要的基础内容之一。它们让你能够通过按逻辑顺序使用逆运算来求出未知数的值。在本文中，我们将学习如何解一步、两步和多步线性方程，然后把这些方法拓展到一元一次不等式。
You will also learn how to translate word problems into equations, how to check your answers, and how to avoid common mistakes. The methods shown here follow the Cambridge Lower Secondary approach and are essential for later topics such as graphs, sequences, and simultaneous equations.
你还将学习如何把文字题转化为方程、如何检验答案以及如何避免常见错误。这里展示的方法遵循剑桥初中数学的教学思路，对于后续的图像、数列和联立方程等主题也至关重要。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is an equation in which the unknown variable appears only to the power of 1. This means you will never see x², x³, or any higher power inside a linear equation.
线性方程是未知数的指数只为 1 的方程。这意味着在线性方程中你不会看到 x²、x³ 或任何更高的幂次。
linear equation has the form ax + b = c, where a, b, and c are numbers and a is not zero.
在 KS3 阶段，我们通常只处理一个未知数，常用 x、y 或 n 表示。一个典型的一元一次方程形式为 ax + b = c，其中 a、b、c 是数字，并且 a 不等于 0。
ax + b = c, a ≠ 0
en though some of these look different, they can all be rearranged into the standard form.
例如 3x + 4 = 19、5n − 7 = 18 和 2(x + 3) = 14。尽管它们看起来不同，但都可以化为标准形式。
The word 'linear' comes from the fact that the equation represents a straight line when drawn on a graph. In one variable, the solution is a single point on the number line.
“线性”一词来源于该方程在图像上表示一条直线。在一元情况下，方程的解是数轴上的一个点。
2. Balancing Method for One-Step Equations | 一步方程的天平法
Think of an equation as a balance scale. The left side and the right side must always have the same value. If you add, subtract, multiply, or divide one side, you must do exactly the same to the other side to keep the scale balanced.
把方程想象成一个天平。左边和右边必须始终保持相等的值。如果你对一边进行加、减、乘、除，你必须对另一边做完全相同的运算，才能保持天平平衡。
te x on one side. The inverse of adding 6 is subtracting 6, so subtract 6 from both sides.
要解 x + 6 = 15，你需要把 x 单独放在一边。加 6 的逆运算是减 6，所以两边同时减去 6。
x + 6 = 15 → x = 15 − 6 → x = 9
lication, use division as the inverse operation. To solve 3x = 21, divide both sides by 3.
对于乘法，用除法作为逆运算。要解 3x = 21，两边同时除以 3。
3x = 21 → x = 21 ÷ 3 → x = 7
Always write one operation per line when solving an equation. This makes your working clear and helps you avoid missing a step or making a sign error.
解方程时每行只写一个运算。这样你的解题过程会更清晰，也能帮助你避免遗漏步骤或出现符号错误。
3. Two-Step Equations | 两步方程
ifferent operations, such as multiplication and addition. A common example is 5x + 2 = 17.
两步方程涉及两种不同的运算，例如乘法和加法。一个常见的例子是 5x + 2 = 17。
rst multiplied by 5 and then 2 is added. To undo this, subtract 2 first, then divide by 5.
解方程时要逆向使用运算顺序。在 5x + 2 = 17 中，x 先乘以 5，然后加上 2。要逆运算，先减去 2，再除以 5。
5x + 2 = 17 → 5x = 17 − 2 → 5x = 15 → x = 15 ÷ 5 → x = 3
rule applies to subtraction. For 4x − 9 = 11, add 9 to both sides first, then divide by 4.
同样的规则也适用于减法。对于 4x − 9 = 11，先两边加 9，再除以 4。
4x − 9 = 11 → 4x = 11 + 9 → 4x = 20 → x = 20 ÷ 4 → x = 5
Remember: addition and subtraction are undone before multiplication and division. This is the opposite of the BIDMAS order used for evaluating expressions.
记住：加减法要在乘除法之前处理。这与计算代数式时使用的 BIDMAS 顺序相反。
4. Equations with Variables on Both Sides | 两边都有未知数的方程
of the equation. For example, 7x − 3 = 4x + 9 has x terms on both the left and the right.
有时未知数出现在方程的两边。例如，7x − 3 = 4x + 9 的左边和右边都含有 x 项。
To solve, collect all variable terms on one side and all constant terms on the other side. A good method is to subtract the smaller x term from both sides first.
解这类方程时，把所有的未知数项移到一边，把所有的常数项移到另一边。一个常用的方法是先把较小的 x 项从两边减去。
, subtract 4x from both sides to get 3x − 3 = 9. Then add 3 to both sides and divide by 3.
在 7x − 3 = 4x + 9 中，两边同时减去 4x 得到 3x − 3 = 9。然后两边加 3，再除以 3。
7x − 3 = 4x + 9 → 7x − 4x = 9 + 3 → 3x = 12 → x = 4
Be very careful with signs when terms change sides. A term that is added on one side becomes subtracted on the other, and vice versa.
当项移到另一边时，一定要特别注意符号。在一边是加的项，移到另一边就变成减，反之亦然。
5. Expanding Brackets First | 先展开括号
If an equation contains brackets, you usually need to expand them before solving. Expanding means multiplying each term inside the bracket by the number or variable outside.
如果方程含有括号，通常需要先展开括号再求解。展开就是用括号外面的数或变量乘以括号内的每一项。
distributive law to write 3x + 15 = 27. Then subtract 15 from both sides and divide by 3.
对于 3(x + 5) = 27，使用分配律写成 3x + 15 = 27。然后两边减去 15，再除以 3。
3(x + 5) = 27 → 3x + 15 = 27 → 3x = 12 → x = 4
sides, expand both sides first. For example, 2(x + 1) = 3(x − 2) becomes 2x + 2 = 3x − 6.
当两边都有括号时，先展开两边的括号。例如，2(x + 1) = 3(x − 2) 可展开为 2x + 2 = 3x − 6。
2(x + 1) = 3(x − 2) → 2x + 2 = 3x − 6 → 2 + 6 = 3x − 2x → 8 = x
Expanding first reduces the equation to a form you already know how to solve. It also prevents mistakes caused by trying to divide or subtract too early.
先展开可以把方程转化为你已经会解的形式。它还能避免因过早进行除法或减法而导致的错误。
6. Clearing Fractions and Decimals | 去分母与去小数
Equations with fractions can look difficult, but they become easier if you clear the fractions first. To do this, multiply every term on both sides by the lowest common denominator.
含分数的方程可能看起来很难，但如果先去掉分母，就会变得简单。方法是把两边每一项都乘以最小公分母。
For x/3 + 1 = 5, the denominator is 3. Multiply every term by 3 to eliminate the fraction.
对于 x/3 + 1 = 5，分母是 3。把每一项都乘以 3，以消去分数。
3 × (x/3) + 3 × 1 = 3 × 5 → x + 3 = 15 → x = 12
nd the lowest common multiple first. For example, x/2 + x/4 = 9 should be multiplied by 4.
如果有不同的分母，先找最小公倍数。例如，x/2 + x/4 = 9 应该每一项乘以 4。
4 × (x/2) + 4 × (x/4) = 4 × 9 → 2x + x = 36 → 3x = 36 → x = 12
such as 10, 100, or 1000, to make all numbers whole. For 0.2x + 0.5 = 1.7, multiply by 10.
对于小数，把每一项乘以 10 的幂，如 10、100 或 1000，使所有数变成整数。对于 0.2x + 0.5 = 1.7，乘以 10。
10 × 0.2x + 10 × 0.5 = 10 × 1.7 → 2x + 5 = 17 → 2x = 12 → x = 6
Clearing fractions and decimals is a useful skill because it turns harder problems into simple whole-number equations.
去分母和去小数是一项很有用的技能，因为它能把较难的问题转化为简单的整数方程。
7. Introduction to Linear Inequalities | 一元一次不等式入门
An inequality shows a range of possible values rather than one exact value. The solution set of an inequality can contain infinitely many numbers.
不等式表示一个取值范围，而不是一个精确的值。一个不等式的解集可以包含无穷多个数。
The four main symbols are: > means 'greater than', < means 'less than', ≥ means 'greater than or equal to', and ≤ means 'less than or equal to'.
四个主要符号是：> 表示“大于”，< 表示“小于”，≥ 表示“大于或等于”，≤ 表示“小于或等于”。
For example, x > 3 means that x can be 3.1, 4, 10, or any number larger than 3. It does not include 3 itself.
例如，x > 3 表示 x 可以是 3.1、4、10 或任何大于 3 的数。它不包括 3 本身。
s x can be 5, 4.9, 0, −10, or any number less than or equal to 5. The value 5 is included.
相反，x ≤ 5 表示 x 可以是 5、4.9、0、−10 或任何小于等于 5 的数。值 5 包含在内。
x > 3, x < 3, x ≥ 3, x ≤ 3
Writing inequalities is important for representing real-life conditions such as 'you must be at least 12 years old' or 'the temperature is below 0°C'.
写不等式对于表示现实生活中的条件很重要，例如“你必须至少 12 岁”或“温度低于 0°C”。
8. Solving and Graphing Inequalities | 解不等式并画数轴图
Solving linear inequalities uses the same balancing method as solving equations. You can add, subtract, multiply, or divide both sides by the same positive number without changing the direction of the inequality sign.
解一元一次不等式使用与解方程相同的天平法。你可以在两边同时加、减、乘或除以同一个正数，而不改变不等号的方向。
However, if you multiply or divide both sides by a negative number, you must reverse the inequality sign. This is the most common error in inequality work.
但是，如果两边乘以或除以一个负数，你必须反转不等号的方向。这是不等式题目中最常见的错误。
For example, solve −2x < 6. Divide both sides by −2 and reverse the sign to get x > −3.
例如，解 −2x < 6。两边除以 −2 并反转符号，得到 x > −3。
−2x < 6 → x > 6 ÷ (−2) → x > −3
and a closed circle for ≥ or ≤. Then draw an arrow in the direction of the allowed values.
要在数轴上画出不等式，> 或 < 用空心圆表示，≥ 或 ≤ 用实心圆表示。然后朝允许值的方向画箭头。
t 2 and shade to the right. For x ≤ −1, place a closed circle at −1 and shade to the left.
对于 x > 2，在 2 处画空心圆并向右画箭头。对于 x ≤ −1，在 −1 处画实心圆并向左画箭头。
9. Writing Equations from Word Problems | 根据文字题列方程
Many KS3 problems ask you to translate a written sentence into an equation. Start by identifying the unknown quantity and giving it a letter such as x.
KS3 的许多题目要求你把文字描述转化为方程。首先要找出未知量，并用一个字母如 x 表示。
Read the information carefully and break it into small parts. Words like 'more than', 'less than', 'twice', and 'sum' tell you which operation to use.
仔细阅读信息，把它拆成小部分。“多于”“少于”“两倍”“和”等词语告诉你该使用哪种运算。
Example: 'Three more than twice a number is 19' translates to 2x + 3 = 19. The phrase 'twice a number' means 2x, and 'three more than' means add 3.
例如：“一个数的两倍再加 3 等于 19” 可转化为 2x + 3 = 19。“一个数的两倍”表示 2x，“再加 3”表示加 3。
2x + 3 = 19 → 2x = 16 → x = 8
the original sentence. If twice 8 plus 3 equals 19, then 2 × 8 + 3 = 19, which is correct.
解出答案后，检查它是否在原句中成立。如果 8 的两倍加 3 等于 19，那么 2 × 8 + 3 = 19，这是正确的。
Other examples include: 'The sum of a number and 7 is 25' becomes x + 7 = 25, and 'Five less than a number is 12' becomes x − 5 = 12.
其他例子包括：“一个数与 7 的和是 25” 可写成 x + 7 = 25，“一个数减去 5 等于 12” 可写成 x − 5 = 12。
10. Checking Your Solution and Common Mistakes | 检验答案与常见错误
Always substitute your solution back into the original
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