Solving Linear Equations | 解一元线性方程

📚 Solving Linear Equations | 解一元线性方程

Linear equations are the foundation of algebra. In this article, we will learn how to solve linear equations in one variable step by step, understand the balance method, and apply these skills to real-world problems.

线性方程是代数的基础。在这篇文章中，我们将一步一步学习如何解一元线性方程，理解平衡法，并将这些技巧应用于实际问题。

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. For example, 2x + 3 = 7 is a linear equation because x appears without a square or cube. The variable is usually represented by a letter such as x, y, or n.

线性方程是变量最高次数为1的方程。例如，2x + 3 = 7 是一个线性方程，因为 x 没有平方或立方。变量通常用字母表示，如 x、y 或 n。

The general form of a linear equation in one variable is:

一元线性方程的一般形式是：

ax + b = c

where a, b, and c are constants, and a ≠ 0. Solving the equation means finding the value of x that makes the equation true.

其中 a、b、c 是常数，且 a ≠ 0。解方程就是找到使等式成立的 x 的值。

2. The Balance Method | 平衡法

Think of an equation like a balance scale. The equals sign means the two sides are balanced. Whatever you do to one side, you must do the same to the other side. This keeps the equation balanced.

把方程想象成一架天平。等号表示两边是平衡的。你对一边做了什么操作，就必须对另一边做同样的操作。这样才能保持方程平衡。

For example, if we have x + 5 = 9, we can subtract 5 from both sides:

例如，如果我们有 x + 5 = 9，我们可以从两边同时减去5：

x + 5 − 5 = 9 − 5

This gives us x = 4. The solution is x = 4.

这样得到 x = 4。解就是 x = 4。

The key rule is: do the same operation (add, subtract, multiply, or divide) to both sides of the equation.

关键规则是：对方程两边进行相同的运算（加、减、乘、除）。

3. Solving Equations with One Operation | 解只含一步运算的方程

Some equations need only one inverse operation. For example, solve x − 3 = 10. Since 3 is subtracted, we add 3 to both sides:

有些方程只需要一步逆运算。例如，解 x − 3 = 10。因为减去了3，我们两边同时加3：

x − 3 + 3 = 10 + 3

So x = 13. Another example: 4x = 20. Since x is multiplied by 4, we divide both sides by 4:

所以 x = 13。另一个例子：4x = 20。因为 x 乘以4，我们两边同时除以4：

x = 20 ÷ 4 = 5

Always use the inverse operation to isolate the variable.

始终使用逆运算来分离变量。

4. Solving Two-Step Equations | 解两步方程

Many equations require two operations. For example, solve 3x + 2 = 14. First, subtract 2 from both sides:

许多方程需要两步运算。例如，解 3x + 2 = 14。首先，两边同时减去2：

3x = 12

Then divide both sides by 3:

然后两边同时除以3：

x = 4

The order matters: first undo addition or subtraction, then undo multiplication or division.

顺序很重要：先抵消加法或减法，再抵消乘法或除法。

5. Variables on Both Sides | 变量在方程两边

Sometimes the variable appears on both sides of the equation. For example, solve 5x = 2x + 9. To collect like terms, subtract 2x from both sides:

有时变量出现在方程两边。例如，解 5x = 2x + 9。为了合并同类项，两边同时减去2x：

5x − 2x = 9

3x = 9

Then divide by 3 to get x = 3.

然后除以3得到 x = 3。

In general, move the smaller variable term to the other side to avoid negative coefficients.

一般来说，把较小的变量项移到另一边，以避免产生负系数。

6. Equations with Brackets | 含括号的方程

If an equation contains brackets, expand them first. For example, solve 2(x + 4) = 16. Expand the bracket:

如果方程含有括号，先去括号。例如，解 2(x + 4) = 16。展开括号：

2x + 8 = 16

Then solve as usual. Subtract 8 from both sides:

然后像平时一样解。两边同时减去8：

2x = 8

So x = 4. Remember to multiply every term inside the bracket by the number outside.

所以 x = 4。记住要用括号外面的数乘以括号内的每一项。

7. Equations with Fractions | 含分数的方程

When an equation has fractions, you can multiply both sides by the common denominator to eliminate them. For example, solve x/2 + 3 = 7. Multiply every term by 2:

当方程含有分数时，可以两边同时乘以公分母来去分母。例如，解 x/2 + 3 = 7。每一项都乘以2：

x + 6 = 14

Then subtract 6 from both sides to get x = 8.

然后两边减去6得到 x = 8。

Another example: (x + 1)/3 = 2. Multiply both sides by 3:

另一个例子：(x + 1)/3 = 2。两边同时乘以3：

x + 1 = 6

x = 5

This method makes fractional equations much simpler.

这个方法让含分数的方程变得简单很多。

8. Equations with Unknown in the Denominator | 未知数在分母中的方程

If the unknown appears in the denominator, multiply by that denominator. For example, solve 12/x = 4. Multiply both sides by x:

如果未知数出现在分母中，就乘以该分母。例如，解 12/x = 4。两边同时乘以 x：

12 = 4x

Then divide both sides by 4:

然后两边除以4：

x = 3

Always check that your answer does not make the denominator zero.

一定要检查你的答案是否会使分母为零。

9. Word Problems | 应用题

Linear equations are useful for solving word problems. The steps are: read carefully, define a variable, form an equation, solve it, and write the answer in words.

线性方程常用于解决应用题。步骤是：仔细阅读，定义变量，列出方程，解方程，然后用文字写出答案。

Example: A number plus 7 is equal to 15. Let the number be x. Then:

例：一个数加上7等于15。设这个数为 x。那么：

x + 7 = 15

Subtract 7 from both sides: x = 8. So the number is 8.

两边减去7：x = 8。所以这个数是8。

Another example: The sum of 3 consecutive numbers is 36. Let the first number be n. Then n + (n + 1) + (n + 2) = 36. This simplifies to 3n + 3 = 36, so n = 11. The numbers are 11, 12, and 13.

另一个例子：三个连续数的和是36。设第一个数为 n。则 n + (n + 1) + (n + 2) = 36。化简得 3n + 3 = 36，所以 n = 11。这三个数是11、12和13。

10. Checking Solutions | 验证解

After solving an equation, always check your answer by substituting it back into the original equation. For example, solve 2x − 5 = 7. Our solution is x = 6. Substitute x = 6:

解完方程后，一定要把答案代回原方程进行验证。例如，解 2x − 5 = 7。我们的解是 x = 6。代入 x = 6：

2(6) − 5 = 12 − 5 = 7

Since both sides are equal, the solution is correct.

因为两边相等，所以解是正确的。

Checking helps you catch careless mistakes and builds confidence.

验证可以帮助你发现粗心错误，也能增强信心。

11. Common Mistakes | 常见错误

Many students make the same mistakes when solving equations. Here are some to avoid:

很多学生在解方程时会犯同样的错误。以下是一些需要避免的：

Forgetting to perform the same operation on both sides.
Forgetting to multiply every term when expanding brackets.
Misapplying the order of inverse operations.
Making sign errors when moving terms.
Not checking the answer.

忘记对两边进行相同的运算。
去括号时忘记乘以每一项。
逆运算的顺序搞错。
移项时出现符号错误。
没有检查答案。

Being careful with these points will help you solve equations accurately.

注意这些要点能帮助你准确解方程。

12. Practice Questions | 练习题目

Try these questions yourself. Solve each equation and check your answer.

请自己尝试这些题目。解每个方程并检查你的答案。

1. x + 8 = 15

2. 3x = 24

3. 2x + 5 = 13

4. 4(x − 1) = 20

5. 7x = 3x + 12

6. (x − 2)/3 = 4

7. 15/x = 5

8. 3(2x + 1) = 21

Answers: 1. x = 7, 2. x = 8, 3. x = 4, 4. x = 6, 5. x = 3, 6. x = 14, 7. x = 3, 8. x = 3.

答案：1. x = 7，2. x = 8，3. x = 4，4. x = 6，5. x = 3，6. x = 14，7. x = 3，8. x = 3。

Keep practicing, and solving linear equations will become second nature.

不断练习，解线性方程就会成为你的第二天性。

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Solving Linear Equations | 解一元线性方程