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

Earlier, way back in the 9th century, mathematical equations containing certain information used to be written in language form instead of mathematical form. For instance, 6 times the unknown number added with 9 gives 39. Now, it can clearly be seen that writing the equation in language form was very long and sometimes too complex. Hence, with more and more theories, from Babylonian algebra to modern algebraic expressions, equations could be written in easier format mathematically and these are called algebraic equations.

Algebraic Expression

An algebraic expression is one that is a combination of different numbers without specifying them. In other words, the numbers are being represented by different variables joined by some mathematical operators. These types of expressions can be solved by various methods like the hit and trial method, direct method, matrix method, etc. But the simplest way of solving them is by the direct method of solving algebraic expressions. For eg:- a simple algebraic expression can be represented as: 

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

Here, a, b and c are three variables representing three numbers, and all of them are joined by some kind of mathematical operator. So, this is an algebraic expression whose roots/solutions may or may not exist. Now, let’s consider the direct method of solving an algebraic expression by solving the following problem: 

Simplify: .

Solution:

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.

Similar Problems

Question 1: Simplify the algebraic expression: {c(a – b) / (a² – b²)} + c / (a + b)

Solution: 

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)

Question 2: Simplify the algebraic expression: {(a² + 2ab + b²) / (a + b)} – b + {a² / (a + b)}

Solution: 

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)