📚 Mastering Linear Equations and Inequalities | 掌握线性方程与不等式
Linear equations and inequalities form a core part of the Cambridge KS3 mathematics curriculum. They provide essential tools for solving problems across algebra, geometry and real-life contexts. This guide explains the key methods step by step, with clear examples and exam-focused tips to build your confidence.
线性方程和不等式是剑桥 KS3 数学课程的核心内容。它们为代数、几何和实际情境中的问题提供了重要工具。本指南将逐步解释关键方法，配以清晰的例子和考试重点提示，帮助你建立信心。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is an algebraic statement in which the highest power of the unknown is one. It links two expressions with an equals sign, and its graph is always a straight line.
线性方程是未知数的最高次数为 1 的代数等式。它用等号连接两个表达式，其图像总是一条直线。
, 2x + 3 = 11 is a linear equation because the variable x appears only to the first power.
例如，2x + 3 = 11 是线性方程，因为变量 x 只出现一次方。
2x + 3 = 11
The solution of a linear equation is the value of the unknown that makes the equation true. In algebra, we often call this value the root of the equation.
线性方程的解是使等式成立的未知数的值。在代数中，我们通常称这个值为方程的根。
2. The Balancing Method | 平衡法
The balancing method treats an equation like a pair of scales. Whatever operation you perform on one side, you must perform the same operation on the other side to keep the equation balanced.
平衡法把方程看作一架天平。无论你对一边进行什么运算，都必须对另一边进行相同的运算，以保持方程平衡。
This idea is the foundation for solving all linear equations. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side.
这个思想是解所有线性方程的基础。如果你对一边加、减、乘或除以一个数，你必须对另一边做完全相同的运算。
Inverse operations help you isolate the variable. Addition and subtraction are inverse operations, and multiplication and division are inverse operations.
逆运算帮助你分离变量。加法和减法是逆运算，乘法和除法也是逆运算。
3. Solving One-Step Equations | 解一步方程
A one-step equation needs only one inverse operation to find the solution. The key is to identify which operation is being applied to the variable and then reverse it.
一步方程只需一个逆运算即可求解。关键是确定对变量施加了哪种运算，然后将其逆转。
If x + 5 = 12, subtract 5 from both sides. This gives x = 7.
如果 x + 5 = 12，两边同时减去 5。于是得到 x = 7。
x + 5 = 12 ⇒ x = 7
If 3x = 18, divide both sides by 3. The solution is x = 6.
如果 3x = 18，两边同时除以 3。解为 x = 6。
3x = 18 ⇒ x = 6
g it back into the original equation. For example, 3 × 6 = 18, so the solution is correct.
始终将答案代回原方程进行检验。例如，3 × 6 = 18，所以解是正确的。
4. Solving Two-Step Equations | 解两步方程
A two-step equation requires two inverse operations. The standard order is to undo addition or subtraction first, then undo multiplication or division.
两步方程需要两次逆运算。标准顺序是先消去加法或减法，再消去乘法或除法。
For the equation 2x − 4 = 10, first add 4 to both sides. This gives 2x = 14.
对于方程 2x − 4 = 10，先两边加上 4。得到 2x = 14。
2x − 4 = 10 ⇒ 2x = 14
Then divide both sides by 2 to isolate x. The solution is x = 7.
然后两边除以 2 来分离 x。解为 x = 7。
x = 7
The same method works with fractions and negative coefficients. Be careful with signs when adding or subtracting negative terms.
同样的方法适用于分数和负系数。在加减负项时要特别注意符号。
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, you must expand them first before solving. Use the distributive property to multiply the term outside the bracket by each term inside.
当方程含有括号时，必须先展开括号再求解。使用分配律，将括号外的项乘以括号内的每一项。
For example, 3(x + 2) = 15 expands to 3x + 6 = 15.
例如，3(x + 2) = 15 展开后得到 3x + 6 = 15。
3(x + 2) = 15 ⇒ 3x + 6 = 15
Subtract 6 from both sides to get 3x = 9. Then divide by 3 to find x = 3.
两边减去 6，得到 3x = 9。然后除以 3，求出 x = 3。
3x = 9 ⇒ x = 3
out for negative signs outside brackets. For instance, −2(x − 5) = 8 becomes −2x + 10 = 8.
注意括号外的负号。例如，−2(x − 5) = 8 变为 −2x + 10 = 8。
6. Equations with Variables on Both Sides | 变量在两侧的方程
Some linear equations have the unknown on both sides of the equals sign. The first step is to collect all variable terms on one side and all constant terms on the other side.
有些线性方程的未知数出现在等号两侧。第一步是将所有变量项移到一侧，将所有常数项移到另一侧。
For 5x + 2 = 2x + 8, subtract 2x from both sides to obtain 3x + 2 = 8.
对于 5x + 2 = 2x + 8，两边减去 2x，得到 3x + 2 = 8。
5x + 2 = 2x + 8 ⇒ 3x + 2 = 8
Then subtract 2 from both sides: 3x = 6. Finally divide by 3 to get x = 2.
然后两边减去 2：3x = 6。最后除以 3，得到 x = 2。
3x = 6 ⇒ x = 2
Moving terms across the equals sign changes their sign. This is a quick way to rearrange equations, but always think of it as doing the same operation on both sides.
把项移到等号另一侧会改变其符号。这是重新整理方程的快捷方法，但要始终理解为对两边进行相同的运算。
7. Introduction to Inequalities | 不等式入门
that one is greater than or less than the other. The symbols used are >, <, ≥ and ≤.
不等式比较两个表达式，并表明一个大于或小于另一个。使用的符号有 >、<、≥ 和 ≤。
or equal to. Similarly, < means strictly less than, and ≤ means less than or equal to.
符号 > 表示严格大于，而 ≥ 表示大于或等于。类似地，< 表示严格小于，≤ 表示小于或等于。
For example, x > 3 means x can be any number larger than 3, such as 3.1, 4 or 10, but not 3 itself.
例如，x > 3 表示 x 可以是任何大于 3 的数，如 3.1、4 或 10，但不能等于 3。
Inequalities often have infinitely many solutions, so they are usually displayed using number lines or interval notation.
不等式通常有无穷多个解，因此一般用数轴或区间符号来表示。
8. Solving Linear Inequalities | 解一元一次不等式
Solving a linear inequality uses the same steps as solving a linear equation. You isolate the variable by using inverse operations on both sides.
解一元一次不等式使用的步骤与解一元一次方程相同。你通过在两边使用逆运算来分离变量。
For 2x − 3 < 7, first add 3 to both sides to get 2x < 10. Then divide by 2 to find x < 5.
对于 2x − 3 < 7，先两边加 3，得到 2x < 10。然后除以 2，求出 x < 5。
2x − 3 < 7 ⇒ 2x < 10 ⇒ x < 5
There is one very important rule that is different from equations. If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
有一条与方程不同的重要规则。如果你对不等式两边乘以或除以一个负数，必须反转不等号的方向。
For example, −2x < 6. Divide both sides by −2, so the sign changes: x > −3.
例如，−2x < 6。两边除以 −2，因此符号改变：x > −3。
−2x < 6 ⇒ x > −3
Always double-check this rule when negative coefficients appear. A common exam error is forgetting to flip the sign.
当出现负系数时，一定要再次检查这条规则。考试中常见的错误就是忘记反转符号。
9. Representing Inequalities on a Number Line | 在数轴上表示不等式
Number lines give a visual way to show the range of values that satisfy an inequality. An open circle is used for strict inequalities such as > or <.
数轴提供了一种直观表示满足不等式的数值范围的方法。对于 > 或 < 这样的严格不等式，使用空心圆。
nclusive inequalities such as ≥ or ≤. The arrow shows which direction the values continue.
对于 ≥ 或 ≤ 这样的包含不等式，使用实心圆。箭头表示数值延续的方向。
at 2 and shade to the right. For x ≤ −1, draw a closed circle at −1 and shade to the left.
对于 x > 2，在 2 处画空心圆，并向右画阴影。对于 x ≤ −1，在 −1 处画实心圆，并向左画阴影。
Some questions ask you to write an inequality from a given number line. Read the circle type and the arrow direction carefully.
有些题目要求根据给出的数轴写出不等式。请仔细读圆点的类型和箭头的方向。
10. Word Problems and Applications | 文字题与实际应用
Many KS3 exam questions ask you to form an equation or inequality from a written description. Start by defining the unknown with a letter, usually x.
许多 KS3 考试题要求你根据文字描述列出方程或不等式。首先用字母定义未知数，通常用 x。
Translate words such as 'more than', 'less than', 'twice', 'sum' and 'difference' into algebraic operations. Then solve and check the answer in context.
将“多于”、“少于”、“两倍”、“和”和“差”等词语转化为代数运算。然后求解，并结合题意检验答案。
For example, the perimeter of a rectangle is 26 cm. The length is 3 cm more than the width. Let the width be x, so the length is x + 3.
例如，一个长方形的周长是 26 厘米。长比宽多 3 厘米。设宽为 x，那么长为 x + 3。
. Simplify to 4x + 6 = 26, so 4x = 20 and x = 5. The width is 5 cm and the length is 8 cm.
周长方程是 2(x + x + 3) = 26。化简得 4x + 6 = 26，因此 4x = 20，x = 5。宽为 5 厘米，长为 8 厘米。
2(x + x + 3) = 26 ⇒ 4x + 6 = 26 ⇒ x = 5
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always write this rule at the top of your working.
一个常见错误是在乘以或除以负数时忘记反转不等号。始终在你的解题过程上方写下这条规则。
Another mistake is applying an operation to only one side of the equation. Remember that every step must keep the equation balanced.
另一个错误是只对方程的一边进行运算。请记住，每一步都必须保持方程平衡。
Arithmetic errors with negative numbers also cause lost marks. Take extra care when subtracting negative terms or expanding brackets with a minus sign outside.
负数运算错误也会导致失分。在减去负项或展开外面带负号的括号时要格外小心。
| Common error | 如何避免 |
|---|---|
| Forgetting to flip the sign | Highlight negative divisor at once |
| Unbalanced operation | Do the same to both sides |
| Sign error with brackets | Expand carefully and check signs |
Always substitute your final answer back into the original equation or inequality. This quick check can catch many small errors before you finish.
始终将最终答案代回原方程或不等式。这个快速检查可以在你完成之前发现许多小错误。
12. Practice Checklist | 练习检查清单
Before your Cambridge KS3 maths exam, make sure you can solve one-step and two-step equations confidently, including those with negative and fractional coefficients.
在剑桥 KS3 数学考试前，确保你能熟练解一步和两步方程，包括带有负系数和分数系数的方程。
You should also be able to expand brackets, collect variable terms on both sides, solve inequalities and represent them on a number line.
你还应该能够展开括号、合并两侧的变量项、解不等式并在数轴上表示它们。
- Solve one-step equations using inverse operations
- 用逆运算解一步方程
- Solve two-step equations in the correct order
- 按正确顺序解两步方程
- Expand brackets before solving
- 先展开括号再求解
- Collect variables on one side
- 将变量移到同一侧
- Reverse the inequality sign when dividing by a negative
- 除以负数时反转不等号
- Draw open and closed circles correctly
- 正确画出空心圆和实心圆
Working through past paper questions and checking each step against the balancing method is one of the best ways to prepare.
练习历年真题，并按照平衡法逐步检查，是备考的最佳方式之一。
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