Question

Solve the given linear equation $\dfrac{{x + 6}}{4} - \dfrac{{5x - 4}}{8} + \dfrac{{x - 3}}{5} = 0$

Answer 1

Hint: Here, we will first take the LCM of the denominators and solve the numerator to form a linear equation with one variable. Solving further, we will be able to find the required value of $x$ which will be our required answer.

Complete step-by-step answer:
In order to solve $\dfrac{{x + 6}}{4} - \dfrac{{5x - 4}}{8} + \dfrac{{x - 3}}{5} = 0$
First of all, we will find out the LCM of the denominators.
As we can see in the denominator we have the numbers 4, 8 and 5.
Hence, in 4 and 8 (as they are common factors), 8 is the LCM.
Thus, we will find the LCM of 8 and 5 where there is no common factor and we will have to multiply both of them to find the LCM.
Therefore, LCM $ = 8 \times 5 = 40$
Hence, LCM of the denominators 4, 8 and 5 is 40.
Now, for the first fraction, in order to make the denominator 40, we will multiply both the numerator as well as the denominator by 10
Similarly, for the second fraction, in order to make the denominator 40, we will multiply both the numerator as well as the denominator by 5
Also, for the third fraction, in order to make the denominator 40, we will multiply both the numerator as well as the denominator by 8
Thus, we get,
$\dfrac{{10\left( {x + 6} \right) - 5\left( {5x - 4} \right) + 8\left( {x - 3} \right)}}{{40}} = 0$
Multiplying both side by 40, we get
$ \Rightarrow 10x + 60 - 25x + 20 + 8x - 24 = 0$
Adding the like terms, we get
$ \Rightarrow - 7x + 56 = 0$
Adding 56 on both sides, we get
$ \Rightarrow 7x = 56$
Dividing both sides by 7, we get
$ \Rightarrow x = 8$
Therefore, the value of $x$ is 8.
Hence, this is the required answer.

Note: We can also check whether our answer is correct or not by substituting the value of $x$ in the given question.
Thus, it is given that:
$\dfrac{{x + 6}}{4} - \dfrac{{5x - 4}}{8} + \dfrac{{x - 3}}{5} = 0$
Here, substituting $x = 8$, we get,
LHS $ = \dfrac{{8 + 6}}{4} - \dfrac{{5\left( 8 \right) - 4}}{8} + \dfrac{{8 - 3}}{5}$
$ \Rightarrow $ LHS $ = \dfrac{{14}}{4} - \dfrac{{36}}{8} + \dfrac{5}{5}$
Solving further,
$ \Rightarrow $ LHS $ = \dfrac{7}{2} - \dfrac{9}{2} + 1$
$ \Rightarrow $ LHS $ = \dfrac{{7 - 9}}{2} + 1$
Now, solving further, we get,
$ \Rightarrow $ LHS $ = \dfrac{{ - 2}}{2} + 1 = - 1 + 1 = 0$
Clearly, LHS $ = $ RHS
Therefore, our answer is correct.
Hence, the required value of $x$ is 8.
Thus, this is the required answer.