G-3 Workbook Content 336: Solving Linear Equations | G-3练习册内容336：解线性方程

📚 G-3 Workbook Content 336: Solving Linear Equations | G-3练习册内容336：解线性方程

Linear equations are the foundation of algebra. In this article, we will explore how to solve simple linear equations step by step, following the style of Workbook Content 336. You will learn the balance method, handling brackets, fractions, variables on both sides, and how to apply these skills to word problems.

线性方程是代数的基础。在本文中，我们将按照练习册内容336的风格，逐步探索如何解简单线性方程。你将学习平衡法、处理括号、分数、两边都有变量的方程，以及如何将这些技能应用到应用题中。

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

A linear equation is an equation where the highest power of the variable is 1. The general form is ax + b = c, where a, b and c are constants, and x is the unknown variable.

线性方程是变量的最高次数为1的方程。一般形式为 ax + b = c，其中 a、b、c 是常数，x 是未知数。

• Example: 2x + 3 = 11 is a linear equation.
• 例子：2x + 3 = 11 是一个线性方程。
• Example: x² + 1 = 5 is not linear, because x is squared.
• 例子：x² + 1 = 5 不是线性的，因为 x 被平方了。

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2. The Balance Method | 平衡法

Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep the balance.

把方程看作一架天平。你对一边做了什么操作，就必须对另一边做同样的操作，以保持平衡。

For example, to solve x + 3 = 8, subtract 3 from both sides:

例如，解 x + 3 = 8，两边都减去 3：

x + 3 − 3 = 8 − 3

Which gives x = 5.

得到 x = 5。

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3. Solving ax = b | 解 ax = b

When the variable is multiplied by a number, divide both sides by that number.

当变量乘以一个数时，两边同时除以这个数。

Solve 4x = 20:

解 4x = 20：

4x ÷ 4 = 20 ÷ 4

So x = 5.

所以 x = 5。

• Check: 4 × 5 = 20 ✓
• 检验：4 × 5 = 20 ✓

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4. Solving x/a = b | 解 x/a = b

If the variable is divided by a number, multiply both sides by that number.

如果变量除以一个数，两边同时乘以这个数。

Solve x/3 = 7:

解 x/3 = 7：

(x/3) × 3 = 7 × 3

So x = 21.

所以 x = 21。

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5. Equations with Brackets | 带括号的方程

First expand the brackets using the distributive law, then solve as usual.

首先使用分配律展开括号，然后像平常一样求解。

Solve 3(x + 2) = 15:

解 3(x + 2) = 15：

3x + 6 = 15

Subtract 6: 3x = 9. Then divide by 3: x = 3.

两边减 6：3x = 9。然后除以 3：x = 3。

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6. Equations with Variables on Both Sides | 变量在两边的方程

Collect the variable terms on one side and constant terms on the other side.

把含变量的项移到一边，常数项移到另一边。

Solve 5x − 4 = 2x + 8:

解 5x − 4 = 2x + 8：

Subtract 2x from both sides: 3x − 4 = 8

两边减 2x：3x − 4 = 8

Add 4: 3x = 12, so x = 4.

加 4：3x = 12，所以 x = 4。

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7. Equations with Fractions | 含分数的方程

Multiply every term by the common denominator to clear fractions.

将所有项乘以公分母以消去分数。

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

解 x/2 + 1 = x/3 + 3：

Multiply by 6: 3x + 6 = 2x + 18

两边乘以 6：3x + 6 = 2x + 18

Then 3x − 2x = 18 − 6, so x = 12.

然后 3x − 2x = 18 − 6，所以 x = 12。

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8. Word Problems | 应用题

Translate the words into an equation, then solve it.

把文字翻译成方程，然后求解。

Example: A number plus 5 is 12. Find the number.

例子：一个数加 5 等于 12。求这个数。

Let x be the number: x + 5 = 12, so x = 7.

设这个数为 x：x + 5 = 12，所以 x = 7。

• Look for keywords: "sum" means +, "product" means ×, "is" means =.
• 注意关键词：“和”表示 +，“积”表示 ×，“是”表示 =。

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9. Common Mistakes | 常见错误

• Forgetting to do the same operation on both sides. | 忘记对两边做同样的运算。
• Sign errors when moving terms. | 移项时符号错误。
• Not multiplying every term when clearing fractions. | 去分母时没有乘以每一项。
• Expanding brackets incorrectly. | 展开括号不正确。

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

始终将答案代回原方程进行检验。

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10. Practice Questions | 练习

1. 2x + 7 = 19

2. 5(x − 3) = 20

3. 3x − 2 = x + 10

4. x/4 + 2 = 6

Answers: 1) x = 6 2) x = 7 3) x = 6 4) x = 16

答案：1) x = 6 2) x = 7 3) x = 6 4) x = 16

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