📚 Solving Linear Equations | 解线性方程
Linear equations are the foundation of algebra in Edexcel IGCSE Mathematics. This article covers the exact techniques you need: balancing, collecting like terms, removing brackets, solving fractional equations and forming equations from word problems.
线性方程是 Edexcel IGCSE 数学中代数部分的基础。本文覆盖考试所需的核心技巧：等式平衡、合并同类项、去括号、解含分数方程以及从应用题列方程。
1. What Is a Linear Equation? | 什么是线性方程？
In a linear equation, the unknown variable only appears to the power 1. It can be written in the general form:
在线性方程中，未知数只以一次方出现。它可以写成一般形式：
ax + b = c (a ≠ 0)
Here x is the unknown, a is the coefficient of x, and b and c are constants. The graph of such an equation is a straight line, which is why it is called 'linear'.
其中 x 是未知数，a 是 x 的系数，b 和 c 是常数。这类方程的图象是一条直线，所以称为“线性”。
2. The Balancing Principle | 天平平衡原理
The most important rule is: whatever you do to one side, you must do to the other side. This keeps the equation balanced, just like a pair of scales.
最重要的规则是：对等号一边做了什么运算，就必须对另一边做完全相同的运算。这样才能保持方程两边平衡，就像一架天平。
For example, to solve x + 5 = 12, subtract 5 from both sides:
例如，解 x + 5 = 12，两边同时减去 5：
x + 5 - 5 = 12 - 5 ⇒ x = 7
Always check your answer by substituting it back into the original equation.
一定要把答案代回原方程检验。
3. One-Step Equations | 一步方程
One-step equations require only one inverse operation. They may involve addition, subtraction, multiplication or division.
一步方程只需要一次逆运算。它们可能涉及加减乘除中的一种运算。
| Type | Equation | Inverse Operation | Solution |
| Subtraction | x - 3 = 7 | Add 3 | x = 10 |
| Addition | x + 6 = 9 | Subtract 6 | x = 3 |
| Multiplication | 4x = 20 | Divide by 4 | x = 5 |
| Division | x/3 = 6 | Multiply by 3 | x = 18 |
Addition and subtraction undo each other; multiplication and division undo each other.
加法与减法互逆；乘法与除法互逆。
4. Two-Step Equations | 两步方程
Two-step equations require two inverse operations, usually one involving addition or subtraction, and one involving multiplication or division.
两步方程需要两次逆运算，通常一次是加减运算，一次是乘除运算。
Solve 3x + 4 = 19:
解 3x + 4 = 19：
3x + 4 - 4 = 19 - 4 ⇒ 3x = 15
3x ÷ 3 = 15 ÷ 3 ⇒ x = 5
Always deal with addition or subtraction first, then multiplication or division.
通常先处理加减，再处理乘除。
5. Collecting Like Terms | 合并同类项
Before solving, simplify the equation by collecting terms that contain x and constant terms separately.
在求解之前，先合并方程中含 x 的项和常数项，对方程进行化简。
Solve 8x + 5 - 3x = 30:
解 8x + 5 - 3x = 30：
(8x - 3x) + 5 = 30 ⇒ 5x + 5 = 30
Subtract 5 from both sides, then divide by 5:
两边先减去 5，再除以 5：
5x = 25 ⇒ x = 5
Be careful to keep the correct sign of each term when rearranging.
移项时要注意保留每项的符号。
6. Unknowns on Both Sides | 方程两边都有未知数
When x appears on both sides, move the smaller x term to the other side to keep the coefficient of x positive.
当 x 出现在等式两边时，通常把较小的 x 项移到另一边，使 x 的系数为正。
Solve 7x - 2 = 4x + 10:
解 7x - 2 = 4x + 10：
7x - 4x = 10 + 2 ⇒ 3x = 12
x = 4
Check: left side = 7(4) - 2 = 26; right side = 4(4) + 10 = 26.
检验：左边 = 7×4 - 2 = 26；右边 = 4×4 + 10 = 26。
7. Expanding Brackets | 去括号
If the equation contains brackets, expand the brackets first, then solve the resulting equation.
如果方程中含有括号，应该先去括号，然后再解化简后的方程。
Solve 3(2x + 5) = 45:
解 3(2x + 5) = 45：
3 × 2x + 3 × 5 = 45 ⇒ 6x + 15 = 45
Subtract 15 from both sides, then divide by 6:
两边减去 15，再除以 6：
6x = 30 ⇒ x = 5
When the number outside the bracket is negative, flip the sign of every term inside the bracket.
当括号外是负数时，括号内每一项的符号都要改变。
8. Equations with Fractions | 含分数方程
To solve an equation with fractions, multiply every term by the common denominator to 'clear' the fractions.
解含分数方程时，两边同时乘以分母的公倍数，从而“去掉”分母。
Solve (x + 2)/3 = 4:
解 (x + 2)/3 = 4：
(x + 2)/3 × 3 = 4 × 3 ⇒ x + 2 = 12
x = 10
For an equation with fractions on both sides, multiply by the lowest common multiple of the denominators.
如果方程两边都有分数，可以乘以所有分母的最小公倍数。
Solve (x + 1)/4 = (x - 3)/2:
解 (x + 1)/4 = (x - 3)/2：
x + 1 = 2(x - 3) ⇒ x + 1 = 2x - 6
1 + 6 = 2x - x ⇒ x = 7
Substitute x = 7 back: 8/4 = 2 and 4/2 = 2, so the solution is correct.
把 x = 7 代回验算：8/4 = 2，4/2 = 2，因此解正确。
9. Forming Equations from Word Problems | 从应用题列方程
To solve word problems, define a variable, translate the English sentence into algebra, then solve the equation.
解应用题时，先设未知数，再把文字条件翻译成代数式，最后解方程。
Example: A rectangle has length 3 cm more than twice its width. The perimeter is 30 cm. Find the width.
例：一个长方形的长比宽的 2 倍还多 3 cm，周长为 30 cm，求宽。
Let the width be w. Then the length is 2w + 3.
设宽为 w，则长为 2w + 3。
Perimeter = 2(w + 2w + 3) = 30
2(3w + 3) = 30 ⇒ 6w + 6 = 30
6w = 24 ⇒ w = 4
he width is 4 cm and the length is 2(4) + 3 = 11 cm. Check: perimeter = 2(4 + 11) = 30 cm.
宽为 4 cm，长为 2×4 + 3 = 11 cm。检验：周长 = 2×(4 + 11) = 30 cm。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
The most common errors in linear equation questions are:
线性方程题中最常见的错误有：
- Dropping the negative sign when moving a term. | 移项时漏掉负号。
- Multiplying only one side when clearing fractions. | 去分母时只乘一边或只乘一项。
- Forgetting to multiply every term inside a bracket by the outside number. | 展开括号时忘记把括号外数字乘到每一项。
- Not substituting the answer back to check. | 没有代回原方程检验。
In the Edexcel IGCSE exam, always show your working clearly. A correct method can still earn marks even if your final answer has a small arithmetic error.
在 Edexcel IGCSE 考试中，务必清晰写出步骤。即使最终答案有轻微计算错误，正确的方法也能得分。
11. Practice Questions | 即时练习
Try these four questions before checking the answers.
先尝试完成下面四道题，再核对答案。
| Question | Answer |
| 2x - 7 = 11 | x = 9 |
| 5(2x + 1) = 45 | x = 4 |
| (3x - 1)/2 = 7 | x = 5 |
| 4x + 3 = 2x + 13 | x = 5 |
Mastering linear equations will help you with inequalities, simultaneous equations and straight-line graphs throughout your IGCSE course.
掌握线性方程，将为你后续学习不等式、联立方程和直线图象打下坚实基础。
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