Question

How do you solve the \[\dfrac{1}{2}(x - 3) = \dfrac{1}{3}(2x + 1)\]?

Answer 1

Hint: In the given problem we need to solve this for ‘x’. We can solve this using the transposition method. The common transposition method is to do the same thing (mathematically) to both sides of the equation, with the aim of bringing like terms together and isolating the variable (or the unknown quantity). That is we group the ‘x’ terms one side and constants on the other side of the equation.

Complete step-by-step solution:
Given, \[\dfrac{1}{2}(x - 3) = \dfrac{1}{3}(2x + 1)\].
Expanding the brackets we have,
\[\Rightarrow \dfrac{x}{2} - \dfrac{3}{2} = \dfrac{{2x}}{3} + \dfrac{1}{3}\]
We transpose \[\dfrac{{2x}}{3}\]to the left hand side of the equation by subtracting \[\dfrac{{2x}}{3}\]on the left hand side of the equation,
\[\Rightarrow \dfrac{x}{2} - \dfrac{{2x}}{3} - \dfrac{3}{2} = \dfrac{1}{3}\]
We transpose \[ - \dfrac{3}{2}\]to the right hand side of the equation by adding \[\dfrac{3}{2}\]on the right hand side of the equation,
\[\Rightarrow \dfrac{x}{2} - \dfrac{{2x}}{3} = \dfrac{1}{3} + \dfrac{3}{2}\]
Taking LCM and simplifying we have,
\[\Rightarrow \dfrac{{3x - 2(2x)}}{6} = \dfrac{{2 + (3 \times 3)}}{6}\]
\[\Rightarrow \dfrac{{3x - 4x}}{6} = \dfrac{{2 + 9}}{6}\]
\[\Rightarrow \dfrac{{ - x}}{6} = \dfrac{{11}}{6}\]
Multiplying -6 on both side of the equation
\[\Rightarrow \dfrac{{ - x}}{6} \times - 6 = \dfrac{{11}}{6} \times - 6\]
\[ \Rightarrow x = - 11\]
This is the required answer.

Note: We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘x’ in the given problem.
\[\Rightarrow \dfrac{1}{2}( - 11 - 3) = \dfrac{1}{3}(2( - 11) + 1)\]
\[\Rightarrow \left( {\dfrac{1}{2} \times - 14} \right) = \dfrac{1}{3}( - 22 + 1)\]
\[ \Rightarrow - 7 = \dfrac{1}{3} \times - 21\]
\[ \Rightarrow - 7 = - 7\]
Hence the obtained answer is correct.

We know that the product of two negative numbers is a positive number. Product of a negative number and a positive number gives negative number (vice versa). If we want to transpose a positive number to the other side of the equation we subtract the same number on that side (vice versa). Similarly if we have multiplication we use division to transpose. If we have division we use multiplication to transpose. Follow the same procedure for these kinds of problems.