Question

Solve the following equation: $ \dfrac{{3x}}{{x + 6}} - \dfrac{x}{{x + 5}} = 2 $

Answer 1

Hint: In order to solve the equation, first solve the left side equation by multiplying the denominators with the alternate numerators, so that we get the common denominator. As we know that if the equations have different denominators which do not have any common factors, then simply multiply the denominator with an alternate numerator. Then simplify the obtained numerator by adding or subtracting to find the value of $ x $ .

Complete step-by-step answer:
We are given with the equation $ \dfrac{{3x}}{{x + 6}} - \dfrac{x}{{x + 5}} = 2 $ .
Multiplying the denominator $ x + 5 $ to the numerator and denominator of the first operand and multiplying the denominator $ x + 6 $ to the numerator and denominator of the second operand, and we get:
 $
  \dfrac{{3x}}{{x + 6}} \times \dfrac{{x + 5}}{{x + 5}} - \dfrac{x}{{x + 5}} \times \dfrac{{x + 6}}{{x + 6}} = 2 \\
  \Rightarrow \dfrac{{3x\left( {x + 5} \right)}}{{\left( {x + 6} \right)\left( {x + 5} \right)}} - \dfrac{{x\left( {x + 6} \right)}}{{\left( {x + 6} \right)\left( {x + 5} \right)}} = 2 \;
  $
And, we got the common denominator.
Since, there is common denominator on both the operands so, we are taking it as one operand and we get:
 $ \dfrac{{3x\left( {x + 5} \right) - x\left( {x + 6} \right)}}{{\left( {x + 6} \right)\left( {x + 5} \right)}} = 2 $
Multiplying both sides by $ \left( {x + 6} \right)\left( {x + 5} \right) $ in order to eliminate the denominator on the left side:
 $
  \dfrac{{\left( {3x\left( {x + 5} \right) - x\left( {x + 6} \right)} \right)}}{{\left( {x + 6} \right)\left( {x + 5} \right)}} \times \left( {x + 6} \right)\left( {x + 5} \right) = 2\left( {x + 6} \right)\left( {x + 5} \right) \\
  \Rightarrow 3x\left( {x + 5} \right) - x\left( {x + 6} \right) = 2\left( {x + 6} \right)\left( {x + 5} \right) \;
  $
Opening the parenthesis of the both the sides and on further solving, we get:
 $
  3x\left( {x + 5} \right) - x\left( {x + 6} \right) = 2\left( {x + 6} \right)\left( {x + 5} \right) \\
  \Rightarrow 3{x^2} + 15x - \left( {{x^2} + 6x} \right) = 2\left( {{x^2} + 6x + 5x + 30} \right) \\
  \Rightarrow 3{x^2} + 15x - {x^2} - 6x = 2\left( {{x^2} + 11x + 30} \right) \\
  \Rightarrow 3{x^2} + 15x - {x^2} - 6x = 2{x^2} + 22x + 60 \\
  \Rightarrow 2{x^2} + 9x = 2{x^2} + 22x + 60 \;
  $
For further simplifying, subtracting both the sides by $ 2{x^2} $ and $ 22x $ :
 $
  \Rightarrow 2{x^2} + 9x = 2{x^2} + 22x + 60 \\
  \Rightarrow 2{x^2} + 9x - 2{x^2} - 22x = 2{x^2} - 2{x^2} + 22x - 22x + 60 \\
  \Rightarrow - 13x = 60 \;
  $
Dividing both the sides by $ - 13 $ , in order to cancel out the constant from the left-hand side, so that we are left with only the variable $ x $ , and on dividing we get:
 $
   - 13x = 60 \\
  \Rightarrow \dfrac{{ - 13x}}{{ - 13}} = \dfrac{{60}}{{ - 13}} \\
  \Rightarrow x = \dfrac{{ - 60}}{{13}} \;
  $
Therefore, after solving the equation $ \dfrac{{3x}}{{x + 6}} - \dfrac{x}{{x + 5}} = 2 $ , we get $ x = \dfrac{{ - 60}}{{13}} $ .
So, the correct answer is “ $ \dfrac{{ - 60}}{{13}} $ ”.

Note: Since, the numerator is greater than the denominator so we can also convert it into the mixed fraction, like $ x = \dfrac{{ - 60}}{{13}} = - 4\dfrac{8}{{13}} $ .
Always preferred to go step by step rather than trying to solve the equation at once, otherwise it leads to error sometimes.