Question

How do you solve \[\dfrac{2}{5}x + \dfrac{1}{4} = - \dfrac{7}{{10}}\] by clearing the fractions ?

Answer 1

Hint: In the given problem we need to solve this for ‘x’. First we need to clear the fraction by taking the LCM of 5, 4 and 10 and multiplying it on both side of the equation. After simplification we will get a simple linear equation. We can solve this using the transposition method. The common transposition method is to do the same thing (mathematically) to both sides of the equation, with the aim of bringing like terms together and isolating the variable (or the unknown quantity). That is we group the ‘x’ terms one side and constants on the other side of the equation.

Complete step by step solution:
Given, \[\dfrac{2}{5}x + \dfrac{1}{4} = - \dfrac{7}{{10}}\] .
We know that the LCM of 5, 4 and 10 is 20.
So multiply 20 on both sides of the equation,
 \[20 \times \dfrac{2}{5}x + 20 \times \dfrac{1}{4} = 20 \times - \dfrac{7}{{10}}\]
 \[\left( {4 \times 2x} \right) + 5 = 2 \times - 7\]
 \[8x + 5 = - 14\]
We transpose \[5\] which is present in the left hand side of the equation to the right hand side of the equation by subtracting ‘5’ on the right hand side of the equation.
 \[8x = - 14 - 5\]
 \[8x = - 19\]
We transpose \[8\] to the right hand side of the equation by dividing \[8\] on the right hand side of the equation.
 \[ \Rightarrow x = \dfrac{{ - 19}}{8}\] . This is the exact form.
 \[ \Rightarrow x = - 2.375\] this is the decimal form.
So, the correct answer is “x = - 2.375”.

Note: We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘x’ in the given problem.
 \[\dfrac{2}{5} \times \dfrac{{ - 19}}{8} + \dfrac{1}{4} = - \dfrac{7}{{10}}\]
 \[
  \dfrac{{ - 19}}{{5 \times 4}} + \dfrac{1}{4} = - \dfrac{7}{{10}} \\
  \dfrac{{ - 19}}{{20}} + \dfrac{1}{4} = - \dfrac{7}{{10}} \\
  \dfrac{{ - 19 + 5}}{{20}} = - \dfrac{7}{{10}} \\
  \dfrac{{ - 14}}{{20}} = - \dfrac{7}{{10}} \\
   \Rightarrow - \dfrac{7}{{10}} = - \dfrac{7}{{10}} \;
\]
Hence the obtained answer is correct.
In the above we did the transpose of addition and subtraction. Similarly if we have multiplication we use division to transpose. If we have division we use multiplication to transpose. Follow the same procedure for these kinds of problems.