Question

Solve the following equation $\dfrac{2x-3}{5}+\dfrac{x+3}{4}=\dfrac{4x+1}{7}$.

Answer 1

Hint: In order to solve this question, we will first convert the fractional terms of the equation to non-fractional terms by taking the LCM and then cross multiplying. After that, we will apply multiple arithmetic operations to get the value of x.

Complete step-by-step solution -
In this question, we have asked to solve the given equation, $\dfrac{2x-3}{5}+\dfrac{x+3}{4}=\dfrac{4x+1}{7}$. To solve this equation, we will first convert them into non fractional terms. So, we will take the LCM of both the terms on the left side of the equality. So, we will get as follows,
$\dfrac{4\left( 2x-3 \right)+5\left( x+3 \right)}{5\times 4}=\dfrac{4x+1}{7}$
Now, we will open the brackets to simplify further. So, we get,
$\dfrac{8x-12+5x+15}{20}=\dfrac{4x+1}{7}$
Now, we know that like terms can be added together. So, we will add 8x and 5x and we will subtract 12 from 15. Therefore, we get,
$\dfrac{13x+3}{20}=\dfrac{4x+1}{7}$
Now, we will cross multiply the terms of the equation. So, we will get as,
$\left( 13x+3 \right)7=\left( 4x+1 \right)20$
Now, we will open the brackets and simplify it further. So, we will get,
$91x+21=80x+20$
Now, we will take the terms with the variable x on to the left hand side and the constants on the right hand side. By doing so, we get,
$\begin{align}
  & 91x-80x=20-21 \\
 & \Rightarrow 11x=-1 \\
\end{align}$
And if we divide the equation by 11, we will get,
$\begin{align}
  & \dfrac{11x}{11}=\dfrac{-1}{11} \\
 & \Rightarrow x=\dfrac{-1}{11} \\
\end{align}$
Hence, the value of x for the equation, $\dfrac{2x-3}{5}+\dfrac{x+3}{4}=\dfrac{4x+1}{7}$ is $\dfrac{-1}{11}$.

Note: We have to be very careful while solving this question as it involves a lot of calculation mistakes and that increases the chances of errors. Also, to solve such types of equations, we try to get x on one side and the other terms on the other side by applying multiple arithmetic operations. We could also have solved it by taking the term $\dfrac{4x+1}{7}$ to the left-hand side in the first step and then taking the LCM of all the three terms and solving further.