Solving Linear Equations and Inequalities | 解线性方程与不等式

📚 Solving Linear Equations and Inequalities | 解线性方程与不等式

Linear equations and inequalities are the foundation of algebra. In this article, we will explore how to solve them step by step, understand their graphical meaning, and apply them to real-world problems.

线性方程与不等式是代数的基础。在本文中，我们将逐步探索如何求解它们，理解它们的图形含义，并将其应用于现实问题。

1. Understanding Variables and Expressions | 理解变量与表达式

In algebra, a variable is a symbol (usually a letter) that represents an unknown number. An expression is a combination of numbers, variables, and operations. For example, \(3x + 5\) is an expression.

在代数中，变量是一个符号（通常是字母），代表一个未知数。表达式是数字、变量和运算的组合。例如，\(3x + 5\) 是一个表达式。

A variable can change its value, while a constant stays the same.

变量可以改变其值，而常数保持不变。
Expressions do not have an equals sign; equations do.

表达式没有等号；方程有等号。

2. The Balance Method | 平衡法

An equation is like a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced.

方程就像一架平衡的天平。你对一边做的操作，必须对另一边也做，才能保持平衡。

To solve an equation, we isolate the variable by performing inverse operations.

为了解方程，我们通过执行逆运算来分离变量。

x + 7 = 12 → x = 12 − 7 → x = 5

3. Solving Two-Step Equations | 解两步方程

Some equations require two operations to solve. The order of operations is reversed: deal with addition/subtraction first, then multiplication/division.

有些方程需要两步运算才能解出。运算顺序要反过来：先处理加减法，再处理乘除法。

2x + 3 = 11 → 2x = 11 − 3 → 2x = 8 → x = 4

Always check your answer by substituting it back into the original equation.

始终通过将答案代回原方程来验算。
Keep the equation balanced by performing the same operation on both sides.

通过在两边执行相同的运算来保持方程平衡。

4. Equations with Brackets | 带括号的方程

When an equation contains brackets, expand them first using the distributive property, then solve.

当方程含有括号时，先用分配律展开括号，然后再求解。

3(x − 2) = 9 → 3x − 6 = 9 → 3x = 15 → x = 5

Be careful with negative signs outside brackets: −2(x + 4) becomes −2x − 8.

注意括号外的负号：−2(x + 4) 变为 −2x − 8。

5. Equations with Fractions | 含分数的方程

To solve equations with fractions, multiply every term by the common denominator to eliminate the fractions.

要解含分数的方程，将每一项乘以公分母以消去分数。

(x/2) + 1 = 3 → (x/2) = 2 → x = 4

Alternatively, multiply the whole equation by the denominator at once.

或者，一次性将整个方程乘以分母。

6. Rearranging Formulas | 重排公式

Rearranging a formula means making a different variable the subject. The rules are the same as solving equations.

重排公式意味着使另一个变量成为被求对象。规则与解方程相同。

For example, make y the subject of x = y + 3: y = x − 3.

例如，在 x = y + 3 中把 y 作为被求对象：y = x − 3。

Treat the formula as an equation, but keep the target variable separate.

将公式视为方程，但让目标变量单独留在一边。
Apply inverse operations in the correct order.

按正确的顺序应用逆运算。

7. Solving Linear Inequalities | 解线性不等式

Inequalities are similar to equations, but they use symbols like <, >, ≤, and ≥. Solving them follows the same rules, except one important twist: when you multiply or divide by a negative number, you must flip the inequality sign.

不等式与方程类似，但使用 <、>、≤ 和 ≥ 等符号。求解规则基本相同，只有一个重要差异：当乘以或除以负数时，必须反转不等号。

−2x < 6 → x > −3

Verification: Try x = 0, then −2(0) = 0 < 6 is true, and 0 > −3 is also true.

验证：尝试 x = 0，则 −2(0) = 0 < 6 成立，且 0 > −3 也成立。

8. Graphing Inequalities on a Number Line | 在数轴上表示不等式

Inequalities can be shown on a number line using open or filled circles. An open circle (○) means the number is not included, while a filled circle (●) means it is included.

不等式可以用数轴上的空心圆或实心圆表示。空心圆（○）表示不包含该数，实心圆（●）表示包含该数。

x > 4: open circle at 4, arrow to the right.

x > 4：在4处画空心圆，向右画箭头。
x ≤ 2: filled circle at 2, arrow to the left.

x ≤ 2：在2处画实心圆，向左画箭头。

9. Word Problems with Equations | 方程应用题

To solve word problems, define a variable, translate the words into an equation, solve it, and then answer the original question.

解应用题时，先定义变量，将文字翻译成方程，求解，然后回答原问题。

Example: A number times 3 plus 5 equals 26. Let n be the number. Then 3n + 5 = 26 → n = 7.

例如：一个数的3倍加5等于26。设这个数为 n。则 3n + 5 = 26 → n = 7。

10. Common Mistakes and Tips | 常见错误与技巧

Forgetting to flip the inequality sign when multiplying/dividing by a negative number.

乘以或除以负数时忘记反转不等号。
Not writing the steps clearly, leading to arithmetic errors.

步骤书写不清晰，导致算术错误。
Always check the solution in the original equation or inequality.

始终将解代回原方程或不等式进行验证。
When solving with fractions, multiply every term by the common denominator, not just the fraction terms.

解含分数方程时，要将每一项乘以公分母，而不仅仅是分数项。

11. Practice Problems | 练习题目

Try these problems to test your understanding:

尝试以下题目来测试你的理解：

Solve 5x − 7 = 18.

解 5x − 7 = 18。
Solve 4(x + 2) = 20.

解 4(x + 2) = 20。
Solve (x/3) + 2 = 5.

解 (x/3) + 2 = 5。
Solve −3x + 4 ≤ 10.

解 −3x + 4 ≤ 10。

12. Summary | 总结

Linear equations and inequalities are essential algebraic tools. By mastering the balance method, expanding brackets, handling fractions, and reversing operations correctly, you can solve almost any linear problem confidently.

线性方程与不等式是必不可少的代数工具。通过掌握平衡法、展开括号、处理分数并正确反转运算，你可以自信地解决几乎任何线性问题。

Remember: practice is the key to success. Keep solving and checking your work!

记住：练习是成功的关键。不断求解并检查你的工作！

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Solving Linear Equations and Inequalities | 解线性方程与不等式