Question

Solve the following linear equation.
 $ \dfrac{n}{2} - \dfrac{{3n}}{4} + \dfrac{{5n}}{6} = 21 $

Answer 1

Hint: To solve the linear equation, we will remove the variable term from the right-hand side if it exists. Usually we put the variable term on the left-hand side and all other (constant) terms on the right-hand side. We will use basic knowledge of mathematical operations like addition, subtraction, multiplication and division.

Complete step-by-step answer:
In this problem, we have a single linear equation in one variable $ n $ . The given equation is $ \dfrac{n}{2} - \dfrac{{3n}}{4} + \dfrac{{5n}}{6} = 21 \cdots \cdots \left( 1 \right) $ . We have to solve the equation $ \left( 1 \right) $ for $ n $ .
Let us simplify the LHS of equation $ \left( 1 \right) $ by taking LCM (least common multiple). Note that here LCM of three numbers $ 2,4,6 $ is $ 12 $ . Therefore, we get
 $
  \dfrac{n}{2} - \dfrac{{3n}}{4} + \dfrac{{5n}}{6} = 21 \\
   \Rightarrow \dfrac{{6n}}{{12}} - \dfrac{{9n}}{{12}} + \dfrac{{10n}}{{12}} = 21 \\
   \Rightarrow \dfrac{{6n - 9n + 10n}}{{12}} = 21 \\
   \Rightarrow \dfrac{{7n}}{{12}} = 21 \cdots \cdots \left( 2 \right) \\
  $
Now we will multiply by the term $ \dfrac{{12}}{7} $ on both sides of equation $ \left( 2 \right) $ . Therefore, we get $
  \dfrac{{7n}}{{12}} \times \dfrac{{12}}{7} = 21 \times \dfrac{{12}}{7} \\
   \Rightarrow n = \dfrac{{21 \times 12}}{7} \\
   \Rightarrow n = 3 \times 12 \\
   \Rightarrow n = 36 \\
  $
Therefore, we can say that $ n = 36 $ is a solution of the given equation.

Note: In this type of problem, we can verify the answer by putting the value of $ n $ in the given equation. Let us put $ n = 36 $ on the LHS of the given equation. Therefore, we get LHS $ \dfrac{{36}}{2} - \dfrac{{3\left( {36} \right)}}{4} + \dfrac{{5\left( {36} \right)}}{6} = 18 - 27 + 30 = 21 $ . Also the RHS of the given equation is $ 21 $ . Hence, the answer is verified. The general form of the linear equation in one variable $ x $ is given by $ ax + b = 0 $ where $ a $ and $ b $ are real numbers. The power of variable $ x $ is $ 1 $ . So, it is called a linear equation. A linear equation in one variable has exactly one solution.