IGCSE Grade 8 Student Book 100: Linear Equations and Inequalities | IGCSE八年级学生用书100：线性方程与不等式

📚 IGCSE Grade 8 Student Book 100: Linear Equations and Inequalities | IGCSE八年级学生用书100：线性方程与不等式

Linear equations are a core topic in IGCSE Mathematics. In this unit, students learn to balance equations, isolate an unknown variable, and solve problems that can be modelled by a straight-line relationship. The ability to work confidently with equations also prepares you for graphs, simultaneous equations, and higher-level algebra tasks.

一元一次方程是 IGCSE 数学的核心专题。在本单元中，学生将学习如何平衡方程、分离未知数，并解决可以用直线关系建模的问题。熟练处理方程还能为学习函数图像、联立方程和更高阶的代数内容打下基础。

1. Understanding Linear Equations | 认识线性方程

nts. The key idea is that the left-hand side (LHS) and the right-hand side (RHS) are equal.

线性方程是指未知数（通常写作 x）只出现一次幂的方程。它的一般形式可以写为 ax + b = c 或 ax + b = cx + d，其中 a、b、c 和 d 是常数。关键思想是方程左边（LHS）和右边（RHS）相等。

ax + b = 0 (a ≠ 0)

2, or 4(x + 3) = 20. You are expected to find the value of x that makes the equation true.

在 IGCSE 考试中，线性方程常以 x + 5 = 12、3x - 7 = 2 或 4(x + 3) = 20 等形式出现。你需要找出使方程成立的 x 值。

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2. Solving One-Step Equations | 解一步方程

To solve a one-step equation, use the inverse operation. If a number is added to x, subtract that number from both sides. If x is multiplied by a number, divide both sides by that number.

解一步方程时，使用逆运算。如果 x 加了一个数，就在方程两边同时减去这个数。如果 x 乘以一个数，就在方程两边同时除以这个数。

x + 5 = 12 → x = 7

3x = 15 → x = 5

Always perform the same operation on both sides. This keeps the equation balanced and follows the rule: what you do to one side, you must do to the other side.

必须始终对方程两边进行相同运算。这样可以保持方程平衡，并遵循规则：对方程一边做什么，对另一边也要做什么。

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3. Solving Two-Step Equations | 解两步方程

A two-step equation involves two operations, such as multiplication and addition. Undo the addition or subtraction first, then undo the multiplication or division.

两步方程涉及两种运算，例如乘法和加法。先逆算加法或减法，再逆算乘法或除法。

2x + 3 = 11

2x = 8

x = 4

4. This order is important because the addition is being undone before the multiplication.

例如，在 2x + 3 = 11 中，两边同时减去 3 得到 2x = 8，然后两边同时除以 2 得到 x = 4。这个顺序很重要，因为在处理乘法之前要先逆算加法。

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4. Equations with Variables on Both Sides | 两边含未知数的方程

When an equation has variable terms on both sides, collect all variable terms on one side and all constant terms on the other side. This is usually done by adding or subtracting the same variable term or constant from both sides.

当方程两边都含有未知数项时，把所有未知数项移到一边，把所有常数项移到另一边。通常通过两边同时加上或减去相同的未知数项或常数来实现。

5x - 4 = 2x + 8

3x = 12

x = 4

x - 4 = 8, then add 4 to both sides to get 3x = 12, and finally divide by 3 to find x = 4.

在例子 5x - 4 = 2x + 8 中，两边同时减去 2x 得到 3x - 4 = 8，然后两边同时加上 4 得到 3x = 12，最后除以 3 得到 x = 4。

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5. Expanding Brackets in Equations | 方程中去括号

operty: a(b + c) = ab + ac. Then simplify and solve using the steps from earlier sections.

当方程中出现括号时，先用分配律去括号：a(b + c) = ab + ac。然后化简，并用前面小节中的步骤求解。

3(x + 2) = 2(x + 5)

3x + 6 = 2x + 10

x = 4

ple, -2(x - 3) becomes -2x + 6 because -2 × -3 = +6. A common mistake is to write -2x - 6.

注意负号。例如，-2(x - 3) 去括号后变为 -2x + 6，因为 -2 × -3 = +6。一个常见错误是写成 -2x - 6。

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6. Solving Linear Inequalities | 解一元一次不等式

Solving a linear inequality is very similar to solving a linear equation. However, when you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign.

解一元一次不等式与解一元一次方程非常相似。但是，当你对不等式两边同时乘以或除以一个负数时，必须改变不等号的方向。

-2x < 6 → x > -3

for greater than or equal to. A strict inequality (< or >) does not include the boundary value.

不等式中使用的符号包括：< 表示小于，> 表示大于，≤ 表示小于或等于，≥ 表示大于或等于。严格不等式（< 或 >）不包括边界值。

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7. Representing Inequalities on a Number Line | 在数轴上表示不等式

Inequalities can be shown on a number line. Use an open circle

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