Question

Solve the linear equation:
\[\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}\]

Answer 1

Hint: First we have to know that the linear equation in one variable is an equation containing a maximum of one variable of order 1 and it is expressed in the form of \[ax + b = 0\], where \[a\] and \[b\] are two integers and \[x\] is a variable. This linear equation has only one solution.
To find the solution of the linear equation in one variable, first clear the fractions using LCM. Then simplify both sides of the equation. After that we have to consider the variable separately to get the solution.

Complete answer:
Given a linear equation in one variable is
\[\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}\]------(1)
In the equation (1), the denominators are \[2,5,3,4\]. Then the LCM of \[2,5,3,4\] is \[60\].
Hence to clear the fractions in the equation (1), we need to multiply by \[60\] throughout the equation (1), we get
\[60 \times \left( {\dfrac{x}{2} - \dfrac{1}{5}} \right) = 60 \times \left( {\dfrac{x}{3} + \dfrac{1}{4}} \right)\]------(2)
Simplifying the equation (2), we get
\[30x - 12 = 20x + 15\]------(3)
Taking the variable \[x\] separately in the equation (3), we get
\[30x - 20x = 15 + 12\]
\[ \Rightarrow \]\[10x = 27\]
\[ \Rightarrow \]\[x = \dfrac{{27}}{{10}}\]
Hence the solution of the equation \[\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}\] is \[x = \dfrac{{27}}{{10}}\].

Note:
Note that Prime factorization is a method which is used to express the given numbers as a product of prime factors. By using the prime factorization method, we can easily find LCM of the given numbers.
To find LCM of given numbers, first find all the prime factors of each given number and express given numbers as a product of prime factors. Then, LCM is the product of the highest powers of all the factors.