Question

Solve the following equation and obtain the value of x
$\dfrac{2x+7}{5}-\dfrac{3x+11}{2}=\dfrac{2x+8}{3}-5$

Answer 1

Hint: In this question, we are given an expression involving fractions, therefore, we should multiply each term by the lcm of the denominators so that the fractions are converted into linear terms. Thereafter, we can take the terms involving x to the left hand side and the other terms to the RHS and divide by an appropriate factor so that only x remains in the LHS. Then, the value of the terms in the RHS will give us the value of x which is the required answer.

Complete step-by-step answer:
The given equation is
$\dfrac{2x+7}{5}-\dfrac{3x+11}{2}=\dfrac{2x+8}{3}-5.......................(1.1)$
As the terms are given in fractions, we should first multiply all the terms by the LCM of the denominators so that the denominators get cancelled out and we get a linear equation. Here the denominators of the terms are 5, 2 and 3 all of which are prime. We know that the LCM of prime numbers is equal to their product. Therefore, We should multiply both side of equation (1.1) by $5\times 2\times 3=30$ to obtain
$\begin{align}
  & 30\times \dfrac{2x+7}{5}-30\times \dfrac{3x+11}{2}=30\times \dfrac{2x+8}{3}-30\times 5 \\
 & \Rightarrow 6\times \left( 2x+7 \right)-15\times \left( 3x+11 \right)=10\times \left( 2x+8 \right)-150 \\
 & \Rightarrow 12x+42-45x-165=20x+80-150.......................(1.2) \\
\end{align}$
Now, in equation (1.2), we should take all the terms involving x to LHS and all the constant terms to RHS to obtain
$\begin{align}
  & 12x+42-45x-165=20x+80-150 \\
 & \Rightarrow -53x=53.......................(1.3) \\
\end{align}$
Now, we should divide both sides of (1.3) by -53 to obtain
$\begin{align}
  & -53x=53 \\
 & \Rightarrow x=-1...............(1.4) \\
\end{align}$
Thus, we obtain the answer to the given question as -1.

Note: We should note that in equation (1.4), we divided both sides by -53 to obtain the value of x. We can divide any non-zero term on both sides of a given equation, however we cannot divide both sides by zero as division by zero is not defined.