Question

Find the value of ‘x’ in \[\dfrac{{2x + 3}}{2} = 5 - \dfrac{{2(x - 4)}}{3}\] .

Answer 1

Hint: We first convert the given problems into proportions. After converting into proportions, using cross product property of proportions we will get the value of ‘x’. On the right hand side we take L.C.M and simplify we get a fraction. If we have \[\dfrac{a}{b} = \dfrac{c}{d}\] then ‘a’ and ‘d’ are called extremes and ‘b’ and ‘c’ is called means.

Complete step-by-step answer:
Given, \[\dfrac{{2x + 3}}{2} = 5 - \dfrac{{2(x - 4)}}{3}\]
In the right hand side of the equation we expand the brackets,
 \[ \Rightarrow \dfrac{{2x + 3}}{2} = 5 - \dfrac{{(2x - 8)}}{3}\]
Taking L.C.M. and simplifying we have,
 \[ \Rightarrow \dfrac{{2x + 3}}{2} = \dfrac{{(5 \times 3) - (2x - 8)}}{3}\]
We know that if a negative number is multiplied by a negative number we get a positive number.
 \[ \Rightarrow \dfrac{{2x + 3}}{2} = \dfrac{{15 - 2x + 8}}{3}\]
 \[ \Rightarrow \dfrac{{2x + 3}}{2} = \dfrac{{15 + 8 - 2x}}{3}\]
Adding in the right hand side of the equations,
 \[ \Rightarrow \dfrac{{2x + 3}}{2} = \dfrac{{23 - 2x}}{3}\]
Thus we have a proportion problem.
Now the cross products property of proportions states that the product of the means is equal to the product of the extremes.
Here means are 2 and \[23 - 2x\] . Extremes are 3 and \[2x + 3\] .
Then we have,
 \[ \Rightarrow 3 \times (2x + 3) = 2 \times (23 - 2x)\]
Multiplying the terms in brackets with the constants,
 \[ \Rightarrow (6x + 9) = (46 - 4x)\]
 \[ \Rightarrow 6x + 9 = 46 - 4x\]
Separating the terms contain ‘x’ and constant,
 \[ \Rightarrow 6x + 4x = 46 - 9\]
Taking ‘x’ as common we have,
 \[ \Rightarrow (6 + 4)x = 37\]
Adding we have,
 \[ \Rightarrow 10x = 37\]
Divide by 10 on both sides we have,
 \[ \Rightarrow x = \dfrac{{37}}{{10}}\]
 \[ \Rightarrow x = 3.7\]
So, the correct answer is “x = 3.7”.

Note: We can check that our obtained answer is correct or not. Substitute the obtained ‘x’ value in the given problem. We get 5.2 = 5.2 hence our obtained answer is correct if not then it’s wrong. Follow the same steps as done above for any problem of proposition for finding the variable. Careful in the calculation part.