Question

How do you solve $\dfrac{3n}{n-1}+\dfrac{6n-9}{n-1}=6$?

Answer 1

Hint: As the given fractions are of the same denominator, so we can directly add the numerators of the fraction. Then we can simplify the obtained equation and find the value of n.

Complete step-by-step solution:
We have been given an equation $\dfrac{3n}{n-1}+\dfrac{6n-9}{n-1}=6$.
We have to solve the given equation.
Now, to solve the given equation first we will solve the fraction.
As the given equation has fractions with the same denominator so we can add the numerators of the fractions. Then we will get
$\begin{align}
  & \Rightarrow \dfrac{3n}{n-1}+\dfrac{6n-9}{n-1}=6 \\
 & \Rightarrow \dfrac{3n+6n-9}{n-1}=6 \\
\end{align}$
Now, multiplying the above obtained equation by $\left( n-1 \right)$ we will get
$\Rightarrow \dfrac{3n+6n-9}{n-1}\times \left( n-1 \right)=6\left( n-1 \right)$
Now, simplifying the above obtained equation we will get
$\Rightarrow 9n-9=6n-6$
Now, the above obtained equation is a linear equation.
Now, by shifting the constant terms to the right side of the equation and keeping the variable terms to the left side of the equation we will get
$\Rightarrow 9n-6n=-6+9$
Now, simplifying the above obtained equation we will get
$\Rightarrow 3n=3$
Dividing the equation by 3 we will get
$\Rightarrow \dfrac{3n}{3}=\dfrac{3}{3}$
Now, simplifying the above obtained equation we will get
$\Rightarrow n=1$
Hence on solving the equation we get the value of n as 1.

Note: The possibility of mistake is that students can solve the equation obtained $\Rightarrow 9n-9=6n-6$ as:
Taking the common factors out we will get
$\Rightarrow 9\left( n-1 \right)=6\left( n-1 \right)$
Now, cancel out the common terms we will get
$\Rightarrow 9=6$
It gives the incorrect answer. Because we have to find the value of variable n and it has disappeared from the equation.