简化 {(2a + b)/(a - b)(a - 2 b)} - {(a + 4 b)/(a - b)(a - 3 b)} - {(a - 7 b)/ (a – 2 b)(a – 3 b)}

早在 9 世纪，包含某些信息的数学方程曾经以语言形式而不是数学形式编写。例如，未知数的 6 乘以 9 得到 39。现在，可以清楚地看到，用语言形式写方程很长，有时也太复杂了。因此，随着越来越多的理论，从巴比伦代数到现代代数表达式，方程可以用更简单的数学格式编写，这些被称为代数方程。

代数表达式

代数表达式是不同数字的组合而不指定它们。换句话说，这些数字由一些数学运算符连接的不同变量表示。这些类型的表达式可以通过多种方法求解，如hit and trial法、直接法、矩阵法等。但最简单的求解方法是求解代数表达式的直接法。例如：- 一个简单的代数表达式可以表示为：

(a + b – c) / (10 × (a + b + c + 1)) = 0

这里，a、b 和 c 是代表三个数字的三个变量，它们都由某种数学运算符连接起来。因此，这是一个代数表达式，其根/解可能存在也可能不存在。现在，让我们考虑通过解决以下问题来解决代数表达式的直接方法：

简化： $\frac{(2a + b}{(a - b)(a - 2b)} - \frac{a + 4b}{(a - b)(a - 3b)} - \frac{a - 7b}{(a - 2b)(a - 3b)}$ .

解决方案：

Now, to simplify the given algebraic expression. This can be easily done by the direct method. 

Step 1: Take LCM which is (a – b)(a – 2b)(a – 3b). So the given expression then becomes,

{(2a + b)(a – 3b)} – {(a + 4b)(a – 2b)} – {(a – 7b)(a – b)} / {(a – b)(a – 2b)(a – 3b)}

Step 2: Solve the numerator part while leaving the denominator part as it is.

{(2a² – 6ab + ab – 3b²) – (a² – 2ab + 4ab – 8b²) – (a² – ab – 7ab + 7b²)} / {(a – b)(a – 2b)(a – 3b)}

Step 3: Simplify the numerator part again,

{(2a² – 3b² – 5ab – a² + 8b² – 2ab – a² – 7b² + 8ab)} / {(a – b)(a – 2b)(a – 3b)}

Step 4: Upon simplifying the numerator part,

(-2b² + ab) / {(a – b)(a – 2b)(a – 3b)}

Step 5: Now, taking b as common from the numerator, we get the expression as:

{b × (a – 2b)} / {(a – b)(a – 2b)(a – 3b)}

Step 6: As it can be clearly seen that (a – 2b) is present in the numerator as well as in the denominator. So, they both cancel each other, and hence, this term can be removed from both numerator and denominator. So, after deleting the term (a – 2b) from both numerator and denominator, the expression is:

b / {(a – b)(a – 3b)}

So, this is the simplest form of the given expression that is obtained by the direct method. Now, this expression can be used wherever needed, and thus, it will become easy to solve other expressions including this.

类似问题

问题1：简化代数表达式：{c(a – b) / (a ² – b ² )} + c / (a + b)

解决方案：

Step 1: The identity (a² – b²) = (a + b)(a – b) is already known. So putting this identity in the given expression,

{c(a – b) / {(a – b)(a + b)}} + c / (a + b)

Step 2: Taking out the common value (a – b) from both numerator and denominator of the first term, 

c/(a + b) + c/(a + b)

Step 3: Since both the terms are the same so directly add them up together and hence the final expression becomes:

2 × c / (a + b)

问题2：简化代数表达式：{(a ² + 2ab + b ² ) / (a + b)} – b + {a ² / (a + b)}

解决方案：

Step 1: The identity (a + b)² = a² + 2ab + b² is already known. So putting this identity in the numerator part of the first term of the given expression, 

{(a + b)² / (a + b)} – b + {a² / (a + b)}

Step 2: In the first term of the expression, take (a + b) as common and so it can be removed from both numerator and denominator. So the expression now becomes,

9(a + b) – b + {a² / (a + b)}

Step 3: Since + b – b becomes 0, so:

a + {a² / (a + b)}

Step 4: Leave the expression here itself or do one more step by taking the LCM and solving further. So, if the LCM is taken further of this expression: 

(2a² + ab) / (a + b)

Step 5: Taking a as common from the numerator part of the expression, the final simplified algebraic expression is: 

a × (2a + b) / (a + b)