Year 9 Unit 4: Linear Equations and Inequalities | 九年级第四单元：线性方程与不等式

📚 Year 9 Unit 4: Linear Equations and Inequalities | 九年级第四单元：线性方程与不等式

Linear equations are the foundation of algebraic problem solving. In this unit you will learn how to isolate an unknown variable, handle brackets, fractions and inequalities, and apply these skills to real-world problems.

线性方程是代数问题求解的基础。本单元将学习如何隔离未知变量，处理括号、分数和不等式，并将这些技能应用到实际问题中。

1. What Is a Linear Equation? | 什么是线性方程？

A linear equation is an equation in which the highest power of the unknown is 1. This means the graph of a linear equation is always a straight line.

线性方程是指未知数的最高次数为 1 的方程。这意味着线性方程的图像总是一条直线。

The general form is ax + b = 0, where a and b are constants and a ≠ 0. The unknown is usually written as x, but any letter can be used.

一般形式为 ax + b = 0，其中 a 和 b 是常数，且 a ≠ 0。未知数通常写作 x，但也可以使用任何字母。

2x + 3 = 11

Consider the equation 2x + 3 = 11. The variable x is raised only to the power 1, so the equation is linear.

以方程 2x + 3 = 11 为例。变量 x 的指数只有 1，因此该方程是线性方程。

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2. The Balancing Method | 天平法

Think of an equation as a balance scale. If you add, subtract, multiply or divide one side by a number, you must do the same to the other side to keep the balance.

可以将方程想象成一台天平。如果对一边进行加、减、乘或除一个数的运算，必须对另一边做同样的运算，才能保持天平平衡。

For example, to solve x + 5 = 12, subtract 5 from both sides. The left side becomes x, and the right side becomes 7.

例如，要解 x + 5 = 12，两边同时减去 5。左边变为 x，右边变为 7。

x + 5 − 5 = 12 − 5 ⇒ x = 7

Always check your answer by substituting it back into the original equation: 7 + 5 = 12, which is true.

始终通过将答案代回原方程进行检验：7 + 5 = 12，成立。

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3. Solving Two-Step Equations | 解两步方程

A two-step equation requires two inverse operations. Usually you undo addition or subtraction first, then undo multiplication or division.

两步方程需要进行两次逆运算。通常先消去加法或减法，再消去乘法或除法。

Solve 3x − 4 = 17. First add 4 to both sides to get 3x = 21. Then divide both sides by 3 to get x = 7.

解方程 3x − 4 = 17。首先两边加 4 得到 3x = 21。然后两边除以 3 得到 x = 7。

3x − 4 + 4 = 17 + 4 ⇒ 3x = 21 ⇒ x = 7

The order of inverse operations is the reverse of the order of operations used to build the expression.

逆运算的顺序与构建表达式时所用运算顺序相反。

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4. Removing Brackets | 去括号

When an equation contains brackets, expand them using the distributive law: a(b + c) = ab + ac. Then solve the resulting equation.

当方程含有括号时，使用分配律展开：a(b + c) = ab + ac。然后解所得的方程。

Solve 3(x + 2) = 21. Expand the left side to get 3x + 6 = 21. Subtract 6 from both sides, then divide by 3 to get x = 5.

解方程 3(x + 2) = 21。展开左边得到 3x + 6 = 21。两边减 6，再除以 3，得到 x = 5。

3(x + 2) = 21 ⇒ 3x + 6 = 21 ⇒ 3x = 15 ⇒ x = 5

Be careful with negative signs: −2(x − 4) expands to −2x + 8, not −2x − 8.

注意负号：−2(x − 4) 展开得到 −2x + 8，而不是 −2x − 8。

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5. Variables on Both Sides | 未知数在两边

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