📚 IGCSE Maths: Solving Linear Equations and Inequalities | IGCSE 数学：线性方程与不等式的解法
Linear equations and inequalities form the backbone of IGCSE algebra. They appear in almost every examination paper, both as direct questions and inside word problems, graphical problems and sequences. Being able to isolate the unknown correctly while preserving balance is an essential skill for all candidates.
线性方程和不等式是 IGCSE 代数的基础。它们几乎出现在每一份试卷中，无论是直接考查，还是隐藏在文字题、图像题和数列题里。能够正确地分离未知数并保持等号或不等号平衡，是所有考生必须掌握的核心技能。
1. What Is a Linear Equation? | 什么是线性方程？
r. In one variable, the general form is ax + b = 0, where a and b are constants and a ≠ 0.
线性方程是指未知数的最高次数为 1 的代数等式。它的图像总是一条直线，因此被称为线性方程。在一元情况下，其一般形式为 ax + b = 0，其中 a 和 b 是常数，且 a ≠ 0。
hest power of x is 2. Solving a linear equation means finding the value of the variable that makes the equation true.
例如，2x + 6 = 14 是线性方程，但 x² + 3x = 10 不是线性方程，因为 x 的最高次数是 2。解线性方程就是求出使等式成立的未知数的值。
ax + b = 0 ⇒ x = -b/a
2. Solving One-Step Equations | 解一步方程
act 5 from both sides to obtain x = 7. If 3x = 18, divide both sides by 3 to obtain x = 6.
一步方程只需要一步逆运算即可解出。如果 x + 5 = 12，两边同时减去 5，得到 x = 7。如果 3x = 18，两边同时除以 3，得到 x = 6。
The golden rule is to do exactly the same operation to both sides of the equation. This keeps the equation balanced, just like a set of scales. Never add, subtract, multiply or divide on one side only.
黄金法则是：必须对等式两边做完全相同的运算。这就像天平一样保持方程平衡。绝不能在只有一边进行加、减、乘或除。
x + 5 = 12 ⇒ x = 12 − 5 = 7
3. Solving Multi-Step Equations | 解多步方程
Multi-step equations involve undoing several operations in reverse order. The normal order of operations is BIDMAS, so when solving equations you reverse this order: undo addition or subtraction first, then multiplication or division.
多步方程需要按照相反的顺序逐步消去多个运算。正常运算顺序是 BIDMAS，因此解方程时要逆向进行：先消去加法或减法，再消去乘法或除法。
x = 6. Always show each step clearly because method marks are awarded for correct working.
对于 2x + 7 = 19，先两边减 7，得到 2x = 12。然后两边除以 2，得到 x = 6。务必清晰展示每一步，因为正确的解题过程可以获得步骤分。
2x + 7 = 19 ⇒ 2x = 12 ⇒ x = 6
4. Unknown on Both Sides | 未知数在等号两边
When the variable appears on both sides of the equation, collect all variable terms on one side and all constant terms on the other side. Start by subtracting the smaller variable term from both sides.
当未知数出现在等号两边时，需要把所有含未知数的项移到一边，把所有常数项移到另一边。通常先减去两边中较小的未知数项。
3x − 3 = 9. Then add 3 to both sides to get 3x = 12. Finally divide by 3 to obtain x = 4.
对于 5x − 3 = 2x + 9，两边减去 2x，得到 3x − 3 = 9。然后两边加 3，得到 3x = 12。最后除以 3，得到 x = 4。
Remember that moving a term across the equals sign changes its sign. A positive term becomes negative, and a negative term becomes positive.
请记住，将某一项移过等号时，它的符号会改变。正项变为负项，负项变为正项。
5. Equations with Brackets | 含括号的方程
+ c) = ab + ac. Take care to multiply every term inside the bracket by the factor outside.
如果方程含有括号，必须先展开括号，再合并同类项。这遵循分配律：a(b + c) = ab + ac。注意将括号外的因数乘以括号内的每一项。
2x + 15. Subtract 2x from both sides to get x + 12 = 15, then subtract 12 to obtain x = 3.
对于 3(x + 4) = 2x + 15，先展开左边得到 3x + 12 = 2x + 15。两边减去 2x 得到 x + 12 = 15，再减去 12 得到 x = 3。
3(x + 4) = 2x + 15 ⇒ 3x + 12 = 2x + 15 ⇒ x = 3
6. Equations with Fractions | 含分数的方程
To solve an equation containing fractions, multiply every term on both sides by the lowest common denominator. This clears the fractions and makes the equation easier to solve.
解含分数的方程时，将两边每一项都乘以最小公分母。这样可以消去分母，使方程更容易求解。
d 2 is 6. Multiply every term by 6: 2x + 6 = 3x. Subtract 2x from both sides to get x = 6.
对于 x/3 + 1 = x/2，3 和 2 的最小公分母是 6。将每一项乘以 6：2x + 6 = 3x。两边减去 2x，得到 x = 6。
Always multiply the whole expression on each side by the denominator, not just the fractional parts. This is a common source of mistakes.
一定要把等式两边的整个表达式都乘以公分母，而不是只乘以分数部分。这是常见的错误来源。
7. Inequalities and the Number Line | 不等式与数轴表示
inequality (< or >), while a closed circle represents inclusive inequality (≤ or ≥).
不等式使用 <、>、≤ 或 ≥ 来比较两个表达式。在数轴上，空心圆表示严格不等式（< 或 >），实心圆表示包含该点的不等式（≤ 或 ≥）。
e inequality x ≤ −1 is shown with a closed circle at −1 and an arrow pointing to the left.
例如，x > 2 在数轴上表示为 2 处的空心圆和向右的箭头。不等式 x ≤ −1 表示为 −1 处的实心圆和向左的箭头。
e included or excluded. For 2 ≤ x < 5, the integer solutions are 2, 3 and 4, but not 5.
当题目要求列出整数解时，务必仔细检查端点是否包含。对于 2 ≤ x < 5，整数解为 2、3 和 4，但不包括 5。
8. Solving Linear Inequalities | 解线性不等式
Solving linear inequalities follows the same steps as solving equations. However, there is one crucial difference: when you multiply or divide both sides by a negative number, you must reverse the inequality sign.
解线性不等式的步骤与解方程相同。但有一个关键区别：当两边同时乘以或除以一个负数时，必须改变不等号的方向。
For −2x < 8, divide both sides by −2 and reverse the sign to obtain x > −4. If you forget to reverse the sign, you will get x < −4, which is incorrect.
对于 −2x < 8，两边除以 −2 并反转符号，得到 x > −4。如果忘记反转符号，会得到 x < −4，这是错误的。
−2x < 8 ⇒ x > −4
9. Compound Inequalities | 复合不等式
lt; 2x + 1 ≤ 9. It means that 3 < 2x + 1 and 2x + 1 ≤ 9 are both true at the same time.
复合不等式是一组连续的不等式，例如 3 < 2x + 1 ≤ 9。它表示 3 < 2x + 1 和 2x + 1 ≤ 9 同时成立。
part. Subtract 1 from all parts: 2 < 2x ≤ 8. Then divide all parts by 2: 1 < x ≤ 4.
解复合不等式时，要在所有三部分上同时进行相同的运算。三部分同时减 1：2 < 2x ≤ 8。然后三部分同时除以 2：1 < x ≤ 4。
This means x is greater than 1 and less than or equal to 4. On a number line, it would show an open circle at 1 and a closed circle at 4, with a solid line segment joining them.
这表示 x 大于 1 且小于等于 4。在数轴上，应在 1 处画空心圆，在 4 处画实心圆，并用实线段连接两点。
10. Word Problems Leading to Linear Equations | 由文字题建立线性方程
Word problems must first be translated into algebra before they can be solved. Choose a letter for the unknown, usually x, and write every condition as an expression or equation. Then solve the equation and answer the original question in context.
文字题必须先转化为代数语言，然后才能求解。为未知数选择一个字母，通常用 x，并将每个条件写成表达式或方程。然后解方程，并结合题意回答原问题。
For example, 'three more than twice a number is 15' translates to 2x + 3 = 15. Subtract 3 and divide by 2: x = 6. The unknown number is 6.
例如，“一个数的两倍再加 3 等于 15”可转化为 2x + 3 = 15。减去 3 再除以 2：x = 6。这个未知数是 6。
Always check that your answer makes sense in the original situation. If the problem asks for a length or a number of people, a negative or fractional answer may need to be rejected.
始终检查答案在原始情境中是否合理。如果题目问的是长度或人数，负数或分数答案可能需要舍去。
11. Common Mistakes to Avoid | 常见错误与规避方法
One common mistake is forgetting to reverse the inequality sign when dividing by a negative number. For example, solving −3x < 12 incorrectly gives x < −4, but the correct answer is x > −4.
一个常见错误是在除以负数时忘记反转不等号。例如，解 −3x < 12 时错误地得到 x < −4，但正确答案是 x > −4。
Another common error is multiplying only some terms by the common denominator when clearing fractions. Always multiply every term on both sides, including the ones without a fraction.
另一个常见错误是在去分母时只将部分项乘以公分母。务必乘以两边的每一项，包括没有分母的项。
Writing 5x + 2 = 3x − 6 as 5x − 3x = 6 instead of 5x − 3x = −6 leads to the wrong answer.
移项时的符号错误也很常见。把 5x + 2 = 3x − 6 写成 5x − 3x = 6 而不是 5x − 3x = −6，会导致答案错误。
12. Exam Tips and Final Summary | 考试技巧与总结
. For 2x + 7 = 19, after finding x = 6, substitute to get 2(6) + 7 = 19, which is correct.
始终将最终答案代回原方程进行检验。对于 2x + 7 = 19，求出 x = 6 后，代回得到 2(6) + 7 = 19，等式成立。
Show every step of working clearly, because IGCSE mathematics awards method marks for correct procedures even if the final answer is wrong. For inequalities, sketch the solution set on a number line if the question requests it.
清晰展示每一步计算，因为 IGCSE 数学即使最终答案错误，也会对正确步骤给予过程分。对于不等式，如果题目要求，请在数轴上画出解集。
Practice a mixture of equation types: one-step, multi-step, unknown on both sides, brackets, fractions and word problems. This will build confidence and speed for the examination.
练习各种类型的方程：一步方程、多步方程、未知数在两边、含括号、含分数和文字题。这将增强信心，提高考试速度。
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