Question

Solve the following linear equation.
A) $5x - 6 = 12 - x$
B) \[\dfrac{n}{2} + 1 = 4 - n\]

Answer 1

Hint: For this question first we will take the variables to one side and put the constants on the other side. Then we will solve the value of the constants and variables and we will get our desired answer.

Complete step-by-step answer:
Step 1: Some important points before solving linear equations.
The given questions are linear equations in one variable, as there is only variable x or n that is involved in given equations.
When a number or variable is transposed to one side to the other side of the equations some mathematical changes take place, i.e. changes to and vice versa, and changes to and vice versa.
Bring all the terms of variables to one side and all constant terms to the other side of the equation, this method helps to find the value of the variable.
Step 2: Solve equation (i)
$5x - 6 = 12 - x$
Bring $ - x$ to the left side, and $ - 6$ to the right side.
Thus, after bringing $ - x$ changes to $ + x$ on the left side, and $ - 6$changes to $ + 6$on the right side.
Thus the equation becomes:
$ \Rightarrow 5x + x = 12 + 6$
We know that like terms can be added together.
Thus, $6x = 18$
Now, to find x, there should be no other multiplier with x. therefore, 6 should be transposed to the right side.
Here 6 is multiplied with x, on the right side, it will divide 18. Thus the equation becomes:
$
   \Rightarrow x = \dfrac{{18}}{6} \\
   \Rightarrow x = 3 \\
 $
Hence, the value of x is 3.
Step 3: Solve equation (ii)
\[\dfrac{n}{2} + 1 = 4 - n\]
Bring $ - n$ to the left side, and $ + 1$ to the right side.
Thus, after bringing $ - n$ changes to $ + n$ on the left side, and $ + 1$ changes to $ - 1$ on the right side.
Thus the equation becomes:
\[ \Rightarrow \dfrac{n}{2} + n = 4 - 1\]
Taking L.C.M. on the left side.
\[ \Rightarrow \dfrac{{n + 2n}}{2} = 3\]
We know that like terms can be added together.
Thus, \[\dfrac{{3n}}{2} = 3\]
Here 2 is dividing 3n, on the right side, it will multiply 3. Thus the equation becomes:
\[
   \Rightarrow 3n = 3 \times 2 \\
   \Rightarrow 3n = 6 \\
 \]
Now, to find n, there should be no other multiplier with n. therefore, 3 should be transposed to the right side.
Here 3 is multiplied with n, on the right side, it will divide 6. Thus the equation becomes:
$
   \Rightarrow n = \dfrac{6}{3} \\
   \Rightarrow n = 2 \\
 $

Hence, the value of n is 2.

Note: The value of the variable is called the solution of the equation.
If the value of a variable satisfies the equation, then the variable is the solution of the equation.
By “satisfy”, we meant R.H.S is equal to L.H.S.
A linear equation may have for its solution any rational number.
The importance of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, a combination of currency notes, and so on can be solved using linear equations.