Solving Linear Equations | 解线性方程

📚 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

The 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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Solving Linear Equations | 解线性方程