Question

How do you solve $\dfrac{3x-1}{3}-\dfrac{x-3}{15}=\dfrac{2x+3}{2}$?

Answer 1

Hint: The equation given in the above question has a variable $x$ which has the power of one. This means that the given equation is a linear equation, which in turn means that it will have a single solution. To solve the equation we first have to simplify it by multiplying with the LCM of the numbers present in the denominators, which are $3$, $15$ and $2$. On multiplying, our equation will get free from the fractions and the simplified equation obtained can be solved by using the basic algebraic operations.

Complete step by step solution:
The equation given in the above question is written as
$\Rightarrow \dfrac{3x-1}{3}-\dfrac{x-3}{15}=\dfrac{2x+3}{2}$
We can see that there are fractions present on both the sides of the equation which are making it cumbersome. So let us first simplify it by multiplying it with the LCM of the denominators $3$, $15$ and $2$, which is equal to $30$. Therefore, on multiplying the above equation by $30$ we get
\[\begin{align}
  & \Rightarrow 30\left( \dfrac{3x-1}{3} \right)-30\left( \dfrac{x-3}{15} \right)=30\left( \dfrac{2x+3}{2} \right) \\
 & \Rightarrow 10\left( 3x-1 \right)-2\left( x-3 \right)=15\left( 2x+3 \right) \\
 & \Rightarrow 30x-10-\left( 2x-6 \right)=30x+45 \\
 & \Rightarrow 30x-10-2x+6=30x+45 \\
 & \Rightarrow 28x-4=30x+45 \\
\end{align}\]
So our equation is much simplified now. We subtract $30x$ from both sides to get
$\begin{align}
  & \Rightarrow 28x-4-30x=30x+45-30x \\
 & \Rightarrow -2x-4=45 \\
\end{align}$
Adding $4$ both sides, we get
$\begin{align}
  & \Rightarrow -2x-4+4=45+4 \\
 & \Rightarrow -2x=49 \\
\end{align}$
Finally, on dividing both sides by $-2$ we get
$\begin{align}
  & \Rightarrow \dfrac{-2x}{-2}=\dfrac{49}{-2} \\
 & \Rightarrow x=-24.5 \\
\end{align}$
Hence, the solution of the given equation is $x=-24.5$.

Note: Be careful regarding the change of the sign when a negative sign is outside a bracket. For example, in one of the steps of the above solution, we had the expression \[30x-10-\left( 2x-6 \right)\]. This is a common mistake to incorrectly simplify the expression as \[30x-10-2x-6\] forgetting the fact that minus and minus must be plus. So the expression must be simplified as \[30x-10-2x+6\]. Also, do verify the obtained solution by plugging it back into the given equation and confirm whether LHS is coming equal to RHS.