## Linear Equation Definition

Linear Equation is defined as an algebraic expression that can be written in the form of x+b = 0, having a variable with exponent (power) of 1. There are 2 types of the linear equation – Linear equation in one variable and linear equation in two variables. The graphical representation of a linear equation in one variable is always a straight line horizontally or vertically and the graphical representation of a linear equation in two variables is also a straight line.

## What is a linear equation?

An equation with the highest exponent 1 of the variable and that equation when presented graphically, forms a straight line is called a linear equation.

### Linear equation in one variable:

A linear equation in one variable is defined as an equation that is used to represent and solve for an unknown quantity. A linear equation in one variable is always in the form “ax+b=0” Where a and b are two integers and x is a variable. For example,

If x+5 = 15.

We need to find “y” from this equation i.e. only one variable is present in this equation. This equation has the highest exponent of 1 and there is only one solution of this equation. So this type of equation is known as a linear equation in one variable. Therefore, we can recognize a linear equation by seeing that only one variable is present with an exponent one on it.

### Linear equation in two variables-

Linear equation in two variables is defined as an equation that is used to represent and solve for 2 unknown quantities. A linear equation in two variables is generally in the form “ax+by+c=0”. Where a, b and c are three integers, both a and b are not equal to 0 and x and y are the two variables in it.

A pair of linear equations in two variables can be solved by using different methods like the elimination method, substitution method, graphical method, and cross-multiplication method. The geometrical or graphical representation of a pair of linear equations in two variables is always a straight line. The general form of pair of linear equations in two variables are given as

p_{1}x + q_{1}y + r_{1} = 0

p_{2}x + q_{2}y + r_{2} = 0

In the above equations, p_{1}, p_{2}, q_{1}, q_{2}, r_{1} and r_{2} are the integers and p_{1}² + q_{1}² ≠ 0 and p_{2}² + q_{2}² ≠ 0. Here, p_{1}, p_{2}, q_{1}, q_{2} are known as coefficients of the variables x and y respectively.

### Linear equation formula:

There are several formulas for linear equations in one variable and linear equations in two variables where a line can be defined in an (x,y) plane. The common formulas are:-

- The linear equation formula in general standard form is written as

(px + qy = r)

Where,

- p ≠ 0
- p, q, r are real numbers
- m = -p/q

- The linear equation formula for the vertical line is written as

x = d

- The linear equation formula for a simple slope-intercept form is written as

(y = mx + c)

Where,

- x and y are the two variables
- m is the slope of the line and
- c is the y-intercept

The value of y when x is 0 is called the y-intercept because (0,y) is the point at which the line crosses the y-axis. Henceforth, the linear equation formula is given as: (y = mx + c)

- The linear equation formula for the Horizontal line is written as

y = b

- The linear equation formula for two-point form is written as

{y – y_{1} )= m(x-x_{1})}

Where,

- (x
_{1},y_{1}) and (x_{2},y_{2}) are the two points on the line - m = (y
_{2}-y_{1})/(x_{2}-x_{1})

- x
_{1}≠x_{2}

- The linear equation formula for intercept form is written as

x/x_{0} + y/y_{0} = 1

Where,

- x
_{0}= x-intercept - y
_{0}= y-intercept

The graphical representation of these linear equation formulas are given below