Question

How do you solve $\dfrac{2}{3}\left( {3x - 5} \right) - \dfrac{1}{2}\left( {4x + 3} \right) = \dfrac{1}{4}\left( {2x - 1} \right) - \dfrac{1}{{12}}$?

Answer 1

Hint: In order to solve this question, first we open the brackets and multiply all the terms. Then once we have multiplied the terms, we remove the terms which can be canceled out. Then we simplify the terms by grouping the variables on the left hand side and the constants on the right hand side. Then we solve and simplify further to get our required answer.

Complete step-by-step solution:
We are given a linear equation to solve in this question.
The equation is given as $\dfrac{2}{3}\left( {3x - 5} \right) - \dfrac{1}{2}\left( {4x + 3} \right) = \dfrac{1}{4}\left( {2x - 1} \right) - \dfrac{1}{{12}}$,
Here we will refer to the left side of the ‘equal to’ symbol as the left hand side and right side of the ‘equal to’ symbol as the right hand side.
Now, let us simplify the equation by opening the brackets and multiplying.
$ \Rightarrow \dfrac{{6x}}{3} - \dfrac{{10}}{3} - \dfrac{{4x}}{2} - \dfrac{3}{2} = \dfrac{{2x}}{4} - \dfrac{1}{4} - \dfrac{1}{{12}}$
Let us simplify on the left hand side of the equation:
$ \Rightarrow \dfrac{{\not{{6x}}}}{{\not{3}}} - \dfrac{{10}}{3} - \dfrac{{\not{{4x}}}}{{\not{2}}} - \dfrac{3}{2} = \dfrac{{2x}}{4} - \dfrac{1}{4} - \dfrac{1}{{12}}$
$ \Rightarrow 2x - \dfrac{{10}}{3} - 2x - \dfrac{3}{2} = \dfrac{{2x}}{4} - \dfrac{1}{4} - \dfrac{1}{{12}}$
Now, let us simplify on the right hand side of the equation:
$ \Rightarrow 2x - \dfrac{{10}}{3} - 2x - \dfrac{3}{2} = \dfrac{{\not{{2x}}}}{{\not{4}}} - \dfrac{1}{4} - \dfrac{1}{{12}}$
$ \Rightarrow 2x - \dfrac{{10}}{3} - 2x - \dfrac{3}{2} = \dfrac{x}{2} - \dfrac{1}{4} - \dfrac{1}{{12}}$
On the left hand side we find that $ + 2x$ and $ - 2x$ , cancel out each other. Thus we have:
$ \Rightarrow - \dfrac{{10}}{3} - \dfrac{3}{2} = \dfrac{x}{2} - \dfrac{1}{4} - \dfrac{1}{{12}}$
Now let us keep the constants on the right hand side and the variables on the left hand side:
$ \Rightarrow - \dfrac{x}{2} = \dfrac{{10}}{3} + \dfrac{3}{2} - \dfrac{1}{4} - \dfrac{1}{{12}}$
Let us take the LCM on the right hand side. Thus we find that the factors are: $2 \times 2 \times 3 \times 3 = 36$
Therefore the denominator on the right hand side is $36$
Therefore, $ - \dfrac{x}{2} = \dfrac{{10 \times \left( {12} \right) + 3 \times \left( {18} \right) - 1 \times \left( 9 \right) - 1 \times \left( 3 \right)}}{{36}}$
On simplifying it further, we get:
$ \Rightarrow - \dfrac{x}{2} = \dfrac{{120 + 54 - 9 - 3}}{{36}}$
$ \Rightarrow - \dfrac{x}{2} = \dfrac{{174 - 12}}{{36}} = \dfrac{{162}}{{36}}$
Now multiplying both sides with$ - 2$, we find:
$x = - \dfrac{{\not{{324}}}}{{\not{{36}}}} = - 9$

Therefore the value of x is -9.

Note: These kinds of equations are known as Linear Equations. They are also known as the first order equation as they only have variables raised to the power of one. They are the simplest and the most elemental type of equations. These equations are generally defined for line in coordinate system
We should be cautious about the signs at the time of simplifying the brackets and expanding.