Question

Solve the equation $\dfrac{{2x + 3}}{2} = 5 - \dfrac{{2\left( {x - 4} \right)}}{3}$
(A) $\dfrac{5}{3}$
(B) $\dfrac{{12}}{{29}}$
(C) $\dfrac{{37}}{{10}}$
(D) None of these

Answer 1

Hint: The value of x in $\dfrac{{2x + 3}}{2} = 5 - \dfrac{{2\left( {x - 4} \right)}}{3}$ can be found by using the method of transposition. We will first simplify the equation by opening the brackets and then cross multiply the terms of the equation and simplify the equation to find the value of x. Method of transposition involves doing the exact same mathematical thing on both sides of an equation with the aim of simplification in mind. This method can be used to solve various algebraic equations like the one given in question with ease.

Complete step-by-step solution:
We would use the method of transposition to find the value of x in $\dfrac{{2x + 3}}{2} = 5 - \dfrac{{2\left( {x - 4} \right)}}{3}$.
Method of transposition involves doing the exact same thing on both sides of an equation with the aim of bringing like terms together and isolating the variable or the unknown term in order to simplify the equation and finding the value of the required parameter.
Now, In order to find the value of x, we need to isolate x from the rest of the parameters such as constant terms.
So, we have, $\dfrac{{2x + 3}}{2} = 5 - \dfrac{{2\left( {x - 4} \right)}}{3}$
Taking LCM and simplifying the right side of equation, we get,
$ \Rightarrow \dfrac{{2x + 3}}{2} = \dfrac{{15 - 2\left( {x - 4} \right)}}{3}$
Opening the brackets, we get,
$ \Rightarrow \dfrac{{2x + 3}}{2} = \dfrac{{15 - 2x + 8}}{3}$
Cross multiplying the terms, we get,
$ \Rightarrow 3\left( {2x + 3} \right) = 2\left( {23 - 2x} \right)$
Opening the brackets, we get,
$ \Rightarrow 6x + 9 = 46 - 4x$
Now, isolating the terms consisting x by shifting the terms in the equation. So, we shift all the constant terms to right and terms consisting of variables to the left.
$ \Rightarrow 6x + 4x = 46 - 9$
Simplifying the calculations, we get,
$ \Rightarrow 10x = 37$
Now, dividing both sides by ten, we get,
\[ \Rightarrow x = \dfrac{{37}}{{10}}\]
Now, we know that division by any power of ten is easy as we just have to place a decimal point to the left of the same number of digits as the number of zeroes in the power of ten. So, we get,
\[ \Rightarrow x = 3.7\]
Hence, the value of x in $\dfrac{{2x + 3}}{2} = 5 - \dfrac{{2\left( {x - 4} \right)}}{3}$ is \[x = 3.7\].

Note: If we add, subtract, multiply or divide by the same number on both sides of a given algebraic equation, then both sides will remain equal. We must remember to reverse the signs of the terms while shifting the terms from one side of the equation to the other side.