Question

Solve the linear equation $8x+3=27+2x$

Answer 1

Hint: In the question we are given an equation which is a linear equation in one variable. To understand this we will first learn definitions and examples of variables, constants, degree of an equation, linear equation. Then we will learn the transposition method and we will use this method to solve our given equation.

Complete step-by-step solution:
In the question we are given a linear equation in one variable.
Variable:
 Variable is a symbol whose value is not fixed means its value can vary. We usually denotes variable with alphabets like $x,y,z$ and so on
Constant:
Constant is a value whose value is fixed means we cannot change its value
Example: $1,2,5,77,100$
Equation:
An Equation is a mathematical statement which consists of variables and constants.
An equation can consist of more than one variable.
Example:
$x+5y=48$ $......eq\left( 1 \right)$
$4y=85$ $......eq\left( 2 \right)$
${{a}^{2}}=81$ $......eq\left( 3 \right)$
In $eq\left( 1 \right)$ we have two variables $x\And y$and $48$ is constant, in $eq\left( 2 \right)$ we have only one variable$y$and $45$ is constant and in $eq\left( 3 \right)$ we have only one variable $a$ and $81$ is constant.
Degree of an equation:
Degree of an equation is the highest power of the variable in the equation.
Example:
${{x}^{8}}+{{x}^{4}}+55=88$
In this the variable is $x$ and highest power of $x$ is $8$ , so degree of our given equation is $8$
Linear equation:
Linear equation is an equation whose degree is $1$ or we can say that whose highest power of variable is $1$ .
Example:
$\begin{align}
  & a+b+c=8......eq\left( 1 \right) \\
 & y+5=7......eq\left( 2 \right) \\
\end{align}$
These both are linear equations where $eq\left( 1 \right)$ is in three variables $a,b\And c$ and $eq\left( 2 \right)$ is in one variable $y$ .
Linear equation in one variable:
Linear equation is an equation whose degree is $1$ and consists of one variable only.
Example:
$\begin{align}
  & x+5=8 \\
 & z+7=52 \\
\end{align}$
These two equations are linear equations in one variable
Basic method to solve linear equation in one variable is Transposition
In transposition method
We shift or transpose various operations from right hand side to left hand side or from left hand side to right hand side.
$+$ Will transpose into $-$
$-$ Will transpose into $+$
$\times $ Will transpose into $\div $
$\div $ Will transpose into $\times $
Now we learn basic terminology which is required for this question.
Now we will proceed to our question.
In the question we are given a linear equation in one variable.
Given equation is $8x+3=27+2x$
First we will try to keep all variable terms on the left hand side and all the constant terms on the right hand side.
So for this we will transpose $2x$ to the right hand side and $3$ to the left hand side.
Now our equation becomes,
$\begin{align}
  & 8x+3=27+2x \\
 & \Rightarrow 8x-2x=27-3 \\
\end{align}$
Now we will solve left hand side and right hand side separately
$\begin{align}
  & 8x-2x=27-3 \\
 & \Rightarrow 6x=24 \\
\end{align}$
Now we will transpose $6$ to right hand side
$\begin{align}
  & 6x=24 \\
 & \Rightarrow x=\dfrac{24}{6} \\
\end{align}$
Solving this we will get
$\begin{align}
  & x=\dfrac{24}{6} \\
 & \Rightarrow x=4 \\
\end{align}$
Hence $x=4$ is our required solution.

Note: Linear equation can be of two or three or more variables. To solve linear equations of two variables we have four methods named as graphical method, substitution method, elimination method, cross multiplication method. To solve linear equations in two variables we need at least two equations of the same variable.