IGCSE Maths: Solving Linear Equations and Inequalities | IGCSE 数学：一次方程与不等式的解法

📚 IGCSE Maths: Solving Linear Equations and Inequalities | IGCSE 数学：一次方程与不等式的解法

Linear equations and inequalities are core algebra skills in IGCSE Mathematics. You will meet them in Paper 2 and Paper 4, often hidden inside word problems, sequences, functions, and coordinate geometry questions. Mastering these techniques gives you reliable marks and builds the foundation for harder algebra topics such as quadratics and simultaneous equations.

一次方程与不等式是 IGCSE 数学中的核心代数技能。它们常出现在试卷二和试卷四中，并且经常隐藏在文字题、数列、函数和坐标几何题里。掌握这些方法可以帮你稳定拿分，也为二次方程、联立方程等更难的代数内容打下基础。

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1. What is a linear equation? | 什么是一次方程？

A linear equation is an equation in which the unknown, usually written as x, is raised only to the power 1. Its general form is ax + b = c, where a, b, and c are constants and a ≠ 0.

一次方程是指未知数（通常写成 x）的最高次数为 1 的方程。它的一般形式是 ax + b = c，其中 a、b、c 是常数，且 a ≠ 0。

For example, 2x + 3 = 11, 5 − x = 1, and 4x = 20 are all linear equations. However, x² + 2 = 6 is not linear because the power of x is 2, and 1/x + 3 = 7 is not linear because x appears in the denominator.

例如，2x + 3 = 11、5 − x = 1 和 4x = 20 都是一次方程。但是 x² + 2 = 6 不是一次方程，因为 x 的次数是 2；1/x + 3 = 7 也不是一次方程，因为 x 出现在分母中。

In IGCSE questions, linear equations often appear after you simplify an expression, expand brackets, or clear fractions. Identifying whether an equation is linear helps you choose the correct method.

在 IGCSE 题目中，一次方程经常是在你化简表达式、去括号或消去分数之后出现的。判断一个方程是否为一次方程，有助于你选择正确的解法。

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2. Balancing the equation: inverse operations | 等式平衡：逆运算

To solve a linear equation, keep the equation balanced by performing the same operation on both sides. The goal is to isolate x on one side. Use inverse operations in reverse BIDMAS order: deal with addition or subtraction first, then multiplication or division.

解一次方程时，要对等式两边同时进行相同的运算，以保持平衡。目标是让 x 单独在等式的一边。按照 BIDMAS 的逆顺序使用逆运算：先处理加减，再处理乘除。

Solve 3x + 5 = 17:

解方程 3x + 5 = 17：

3x + 5 − 5 = 17 − 5
3x = 12
x = 12 ÷ 3 = 4

Always substitute your answer back into the original equation to check it. Here, 3(4) + 5 = 12 + 5 = 17, so the solution is correct.

一定要把答案代回原方程进行检验。这里 3(4) + 5 = 12 + 5 = 17，所以答案正确。

When x has a negative coefficient, add or subtract the x term to move it to the other side. For example, 10 − 2x = 4 becomes 10 − 4 = 2x, giving 6 = 2x and x = 3.

当 x 的系数为负数时，可以通过加减 x 项把它移到另一边。例如 10 − 2x = 4 可以变成 10 − 4 = 2x，得到 6 = 2x，所以 x = 3。

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3. Solving equations with brackets | 解含括号的方程

If an equation contains brackets, expand them first. Multiply each term inside the bracket by the number or expression outside, then collect like terms before using inverse operations.

如果方程中含有括号，要先去括号。用括号外的数或表达式乘以括号内的每一项，然后合并同类项，再使用逆运算求解。

Example 1: Solve 2(x + 3) = x + 10.

例题 1：解方程 2(x + 3) = x + 10。

2x + 6 = x + 10
2x − x = 10 − 6
x = 4

Example 2: Solve 3 − 2(x − 1) = 5.

例题 2：解方程 3 − 2(x − 1) = 5。

3 − 2x + 2 = 5
5 − 2x = 5
−2x = 0
x = 0

Be careful with negative signs outside brackets. Multiplying −2 by (x − 1) gives −2x + 2, not −2x − 2. This is one of the most common algebra errors in IGCSE papers.

注意括号外是负号的情况。用 −2 乘以 (x − 1) 得到 −2x + 2，而不是 −2x − 2。这是 IGCSE 试卷中最常见的代数错误之一。

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4. Solving equations with fractions | 解含分数的方程

To solve an equation containing fractions, multiply every term on both sides by the lowest common denominator (LCD). This clears the fractions and produces a simpler linear equation.

解含有分数的方程时，要把两边每一项都乘以最小公分母（LCD）。这样可以消去分母，得到一个更简单的一次方程。

Example 1: Solve x/3 + 1 = x/2.

例题 1：解方程 x/3 + 1 = x/2。

The LCD of 3 and 2 is 6. Multiply every term by 6:

3 和 2 的最小公分母是 6。每一项乘以 6：

6(x/3) + 6(1) = 6(x/2)
2x + 6 = 3x
6 = x

Check: 6/3 + 1 = 2 + 1 = 3 and 6/2 = 3. Both sides match, so x = 6 is correct.

检验：6/3 + 1 = 2 + 1 = 3，6/2 = 3。两边相等，所以 x = 6 正确。

Example 2: Solve (2x − 1)/3 = 5.

例题 2：解方程 (2x − 1)/3 = 5。

3 × (2x − 1)/3 = 3 × 5
2x − 1 = 15
2x = 16
x = 8

When the numerator has more than one term, put it in brackets before multiplying, as shown in Example 2. This helps you avoid the mistake of only multiplying part of the numerator.

当分子不止一项时，在乘法前先把分子用括号括起来，如例题 2 所示。这样可以避免只乘分子的一部分而出错。

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5. Linear inequalities and notation | 一次不等式及其符号

An inequality compares two expressions using <, >, ≤, or ≥. The solution to a linear inequality is usually a range of values rather than a single number.

不等式使用 <、>、≤ 或 ≥ 来比较两个表达式。一次不等式的解通常是一个数值范围，而不是一个数。

Symbol 符号	Meaning 含义	Example 例子
<	less than 小于	x < 3
>	greater than 大于	x > −2
≤	less than or equal to 小于等于	x ≤ 5
≥	greater than or equal to 大于等于	x ≥ 0

Remember: the small end of the symbol points to the smaller value. So x < 3 means x is smaller than 3, and x > 3 means x is greater than 3.

记住：符号的尖角指向较小的数值。所以 x < 3 表示 x 小于 3，x > 3 表示 x 大于 3。

Strict inequalities use < or > and do not include the value. Weak inequalities use ≤ or ≥ and do include the value.

严格不等式使用 < 或 >，不包含边界值。非严格不等式使用 ≤ 或 ≥，包含边界值。

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6. Solving inequalities and reversing the sign | 解不等式与变号规则

Solve inequalities exactly like linear equations, but with one crucial rule: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

解不等式的方法与解一次方程基本相同，但有一条关键规则：如果将不等式两边同时乘以或除以一个负数，必须反转不等号方向。

Example 1: Solve 4 − 3x > 10.

例题 1：解不等式 4 − 3x > 10。

−3x > 10 − 4
−3x > 6
x < 6 ÷ (−3)
x < −2

Notice the sign changed from > to < because we divided by −3. Forgetting to reverse the sign is a very common mistake.

注意，因为除以 −3，不等号从 > 变为 <。忘记反转不等号是一个非常常见的错误。

Example 2: Solve 5 − 2x ≤ 9.

例题 2：解不等式 5 − 2x ≤ 9。

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