Question

How do you simplify $\dfrac{1}{2}\left( {12 + n} \right) + 4n - 3$?

Answer 1

Hint: In order to simplify the above given linear expression in one variable $n$, first expand the terms using the distributive property of multiplication. After this combine all the like terms by resolving all the operators between the like terms. You will get your required result .

Complete step by step solution:
We are given a linear expression in variable $n$ as $\dfrac{1}{2}\left( {12 + n} \right) + 4n - 3$. Let the function be called as $f\left( n \right)$.
$f\left( n \right) = \dfrac{1}{2}\left( {12 + n} \right) + 4n - 3$
So in order to simplify the expression , we first have to expand all the terms and then combine all the like terms obtained .
To expand the first term , use the distributive property of multiplication as $A\left( {B + C} \right) = AB + AC$, we get our expression as
\[
  f\left( n \right) = \dfrac{1}{2}\left( {12} \right) + \dfrac{1}{2}\left( n \right) + 4n - 3 \\
  f\left( n \right) = 6 + \dfrac{n}{2} + 4n - 3 \\
 \]
Hence, we have expanded all the terms successfully.
Now to simplify the expression, we have to combine all the like terms by resolving the arithmetic operations between them as expressed in the expression. Like terms are the terms having the same variable ,so terms with $n$will combine each other and similarly constant terms.
We get our expression as
\[
  f\left( n \right) = 3 + \dfrac{{n + 8n}}{2} \\
  f\left( n \right) = 3 + \dfrac{{9n}}{2} \\
 \]

Therefore, the simplification of the expression $\dfrac{1}{2}\left( {12 + n} \right) + 4n - 3$ is equal to \[3 + \dfrac{{9n}}{2}\].

Additional Information:
Linear Equation: A linear equation is a equation which can be represented in the form of $ax + c$ where $x$ is the unknown variable and a,c are the numbers known where $a \ne 0$. If $a = 0$ then the equation will become constant value and will no more be a linear equation .
The degree of the variable in the linear equation is of the order 1.
Every Linear equation has 1 root.

Note: 1. One must be careful while calculating the answer as calculation error may come.
2. Like terms are terms which have the same variable and same exponent power.
3. Coefficients of the like terms may differ.
4. Proper expansion of all the terms is compulsory for further simplification.