# Linear equations in one variable

## Definition of Linear Equation in one variable

The linear equation which has only one variable is the linear equation in one variable. The general form of linear equation with one variable is:

ax + b = c

The graph of linear equation in one variable is a point in the Cartesian Co-ordinate system.

## Solution of Linear Equations in one variable

Let, a linear equation be, ax + b = c

• Case 1: When c = 0,

The equation can be written as

$ax+ b = 0$

Then, the solution will be:

$ax = - b$

$x = - \frac{b}{a}$

Which is a unique solution.

• Case 2: When, $[c \ne 0]$,

The equation is

$ax + b =c$

The solution of the above equation can be written as:

$ax = c - b$

$x = \frac{{c - b}}{a}$

Which is a unique solution.

• Case 3: When a = 0

If a = 0, then b = c and every number is a solution or it has infinitely many solutions.

If , $[b \ne c]$ then it has no solution, or the equation is said to be inconsistent.

## Example for the solution of linear equation in one variable:

To understand the solution of linear equation in one variable, let us take a linear equation as

x+2=4

The above linear equation has only one variable which is x.

The solution for the above equation means that the value of x for which the above equation is satisfied or the RHS is equal to the LHS.

For finding the value of x for which the equation will be satisfied, let’s transfer the second term of LHS which is 2 to the RHS of the equality sign.

$x - 2 = 4$
$x = 4 - 2$
$x = 2$

Now, let’s check whether x=2 satisfies the equation or not. To do this, we are going to put the x=2 to the LHS of the equation.

$LHS = x + 2$

$= 2 + 2$

$= 4$

By putting x=2, we got 4 in the LHS. The term in the RHS of the equation is also 4.

That means the equation is satisfied or LHS=RHS.

Therefore, x=2 is the solution for the above linear equation having one variable.

Every linear equation which has only one variable has only one solution. That means for only one value of x, the equation will be satisfied. For example, if we put x=4, then the equation will not be satisfied.

Therefore, it may be concluded that any linear in one variable has a unique solution.