# Framing a Linear Equation

## Linear equation in one variable

An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables

Linear equations in one variable, of the type ax + b = 0, can also expressed as a linear equations in two variables. Since, ax + b = 0 ⇒ ax + 0.y + b = 0.

A solution of a linear equation in two variables is a pair of values, one for x and one for y, which satisfy the equation.

• The solution of a linear equation is not effected when
• The same number is added or subtracted from both the sides of an equation.
• Multiplying or dividing both the sides of the equation by the same non-zero number 

When an equation has only one variable of degree one, then that equation is known as linear equation in one variable.

• Standard form: ax+b=0, where a and b ϵ R & a ≠ 0
• Examples of linear equation in one variable are :
• 3x-9 = 0
• 2t = 5

## Linear equation in 2 variables

When an equation has two variables both of degree one, then that equation is known as linear equation in two variables.

Standard form: ax+by+c=0, where a,b,c ϵ R & a,b ≠ 0 Examples of linear equations in two variables are:

• 7x+y=8
• 6p-4q+12=0

# Examples of a Linear Equations

## Solution of linear equation in 2 variables

A linear equation in two variables has infinitely many solutions

Every point on the line satisfies the equation of the line and every solution of the equation is a point on the line.

A linear equation in two variables is represented geometrically by a straight line whose points make up the collection of solutions of the equation. This is called the graph of the linear equation.

x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.

The graph of x = k is a straight line parallel to the y-axis.

For example, the graph of the equation x = 5 is as follows:

A linear equation in two variables has a pair of numbers that can satisfy the equations. This pair of numbers is called as the solution of the linear equation in two variables.

• The solution can be found by assuming the value of one of the variable and then proceed to find the other solution.
• There are infinitely many solutions for a single linear equation in two variables.

The graph of y = k is a straight line parallel to the x-axis.

For example, the graph of the equation y = 5 is as follows:

An equation of the type y = mx represents a line passing through the origin, where m is a real number. For example, the graph of the equation y = 2x is as follows:

# Graph of a Linear Equation

## Graphical representation of a linear equation in 2 variables

• Any linear equation in the standard form ax+by+c=0 has a pair of solutions (x,y), that can be represented in the coordinate plane.
• When an equation is represented graphically, it is a straight line that may or may not cut the coordinate axes.

## Solutions of Linear equation in 2 variables on a graph

• A linear equation ax+by+c=0 is represented graphically as a straight line.
• Every point on the line is a solution for the linear equation.
• Every solution of the linear equation is a point on the line. Lines passing through origin
• Certain linear equations exist such that their solution is (0,0). Such equations when represented graphically pass through the origin.
• The coordinate axes x-axis and y-axis can be represented as y=0 and x=0 respectively. Lines parallel to coordinate axes
• Linear equations of the form y=a, when represented graphically are lines parallel to the x-axis and a is the y-coordinate of the points in that line.
• Linear equations of the form x=a, when represented graphically are lines parallel to the y-axis and a is the x-coordinate of the points in that line.