Question

Solve $ \dfrac{{x - 5}}{2} - \dfrac{{x - 3}}{5} = \dfrac{1}{2} $

Answer 1

Hint: Here we will use the concepts of LCM (least common multiple) to combine the numerator and then will perform cross multiplication to get the simplified equation and then will take the “x” subject moving all the terms in the opposite sides of the equation and simplify for the required value.

Complete step-by-step answer:
Take the given expression: $ \dfrac{{x - 5}}{2} - \dfrac{{x - 3}}{5} = \dfrac{1}{2} $
Take LCM (least common multiple) for the term in the denominator on the left hand side of the equation and then combine the numerator.
 $ \dfrac{{5(x - 5) - 2(x - 3)}}{{2(5)}} = \dfrac{1}{2} $
Open the brackets and multiply the term outside with the terms inside the bracket. When there is a negative sign outside the bracket then the sign of the terms inside the bracket changes when opened. Negative term becomes positive and vice-versa. Also, when there is a positive term outside the bracket then the sign of the term remains the same.
 $ \dfrac{{5x - 25 - 2x + 6}}{{2(5)}} = \dfrac{1}{2} $
Combine like terms and then simplify –
 $ \dfrac{{3x - 19}}{{2(5)}} = \dfrac{1}{2} $
Cross multiply the above expression, where the denominator of one side is multiplied with the numerator of the opposite side and vice versa
 $ \Rightarrow 2(3x - 19) = 2(5) $
Common multiples from both the sides of the equation cancels each other.
 $ \Rightarrow 3x - 19 = 5 $
Make the term with “x” the subject and move constantly on the opposite side. When you move any term from one side to another side of the equation, then the sign of the term also changes. Positive term become negative and vice-versa.
 $ \Rightarrow 3x = 5 + 19 $
Simplify the above expression finding the sum of the terms in the right hand side of the equation –
 $ \Rightarrow 3x = 24 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow x = \dfrac{{24}}{3} $
Common terms from the denominator and the numerator cancel each other.
 $ \Rightarrow x = 8 $
This is the required solution.
So, the correct answer is “x = 8”.

Note: Be careful about the sign convention while expanding the brackets. When there is a negative sign outside the bracket then the sign of the term changes when brackets are opened. While when the sign is positive outside the bracket then the sign of the terms remains the same when brackets are opened.