Question

Solve $5x - 2(2x - 7) = 2(3x - 1) + \dfrac{7}{2}$

Answer 1

Hint: First we will simplify the left-hand and right-hand terms one by one and then we will equate both of them to find the value of $x$.

Complete step-by-step answer:
It is given that the equation $5x - 2(2x - 7) = 2(3x - 1) + \dfrac{7}{2}$
Now we have to find out the Left-Hand Side of the given equation,
$ 5x - 2(2x - 7)$
Upon Multiplying the terms, we get:
$ = 5x - 2 \times 2x - 2 \times 7$, because $2$ is to be multiplied with the entire bracket.
Upon simplifying we get:
$ = 5x - 4x + 14$
Since there are two terms which have $x$ we can subtract them, on subtraction we get:
$ = x + 14$
This is the required simplification for the left-hand side.
Now we have to found the Right-Hand side given equation,
$ = 2(3x - 1) + \dfrac{7}{2}$
On multiplying the terms, we get:
$ = 6x - 2 + \dfrac{7}{2}$
On taking L.C.M (Lowest common multiple) for the fraction we get:
$ = \dfrac{{12x}}{2} - \dfrac{4}{2} + \dfrac{7}{2}$
Since the denominator is same for all the fractions they can be grouped as:
$ = \dfrac{{12x - 4 + 7}}{2}$
On simplifying the numerator, we get:
$ = \dfrac{{12x + 3}}{2}$ , which is the required simplification for the right-hand side.
Now upon equating both the right-hand side and the left-hand side we get:
$ = x + 14 = \dfrac{{12x + 3}}{2}$
On cross multiplying we get:
$ = 2(x + 14) = 12x + 3$
Now upon simplifying we get:
$ = 2x + 28 = 12x + 3$
On taking similar terms across the $ = $ sign we get:
$ = 12x - 2x = 28 - 3$
On simplification we get:
$ = 10x = 25$
Therefore,
$ \Rightarrow x = \dfrac{{25}}{{10}}$
This could be further simplified to:
\[ x = \dfrac{5}{2}\]

The required value of x is $\dfrac{5}{2}$.

Note: To check whether the answer is correct, we will substitute the value of $x$ calculated in the main equation.
The given equation is $5x - 2(2x - 7) = 2(3x - 1) + \dfrac{7}{2}$
First we have to taking Left-hand side:
$ = 5x - 2(2x - 7)$
On substituting the value $x = \dfrac{5}{2}$ we get:
$ = 5\left( {\dfrac{5}{2}} \right) - 2\left( {2\left( {\dfrac{5}{2}} \right) - 7} \right)$
On simplifying we get:
$ = \dfrac{{25}}{2} - 2\left( {\dfrac{{10}}{2} - 7} \right)$
Let us multiply the bracket terms we get
$ = \dfrac{{25}}{2} - \dfrac{{20}}{2} + 14$
On taking L.C.M we get:
$ = \dfrac{{25 - 20 + 28}}{2}$
On adding the terms and we get
$ = \dfrac{{33}}{2}$
Now taking Right-hand side:
$ = 2(3x - 1) + \dfrac{7}{2}$
On substituting the value of $x$ we get:
$ = 2\left( {3\left( {\dfrac{5}{2}} \right) - 1} \right) + \dfrac{7}{2}$
On simplifying we get:
$ = 2\left( {\dfrac{{15}}{2} - 1} \right) + \dfrac{7}{2}$
On multiply the terms and we get
$ = \dfrac{{30}}{2} - 2 + \dfrac{7}{2}$
On taking L.C.M we get:
$ = \dfrac{{30 - 4 + 7}}{2}$
Let us add the terms and we get
$ = \dfrac{{33}}{2}$
Since Left-hand side $ = $ Right-hand side,
Hence, the value of $x = \dfrac{5}{2}$ is correct.