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Graphical Method of Solution

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Graphical Method of Solution - Lesson Summary

Algebraic expression
A combination of constants and variables , connected by four fundamental arithmetical operations of +,-, × and ÷ is called Algebraic expression.

Equation
An algebraic expression with equal sign is called the equation.

Linear equation
A greatest power of the variable in an equation is one then it is called linear equation.

There are many things that share a one-to-one relationship with each other, for example the quantity and cost of things, the age and the height, the altitude and the temperature. Such relationships are linear in nature and can be expressed mathematically as a pair of linear equations in two variables.

Graphical Method of solving pair of linear equations in two variables
The general form of a pair of linear equations in two variables x and y as a 1x + b 1y + c 1 = 0 ,a 2x + b 2y + c 2 = 0 , Where a 1, a 2, b 1, b 2, c 1, c 2 are all real numbers ,a 1 2 + b 1 2 ≠ 0 and a 2 2 + b 2 2 ≠ 0.
 
A linear equation in two variables when plotted on a graph defines a line. So, this means when a pair of linear equations is plotted, two lines are defined. Now, there are two lines in a plane can intersect each other, be parallel to each other, or coincide with each other. The points where the two lines intersect are called the solutions of the pair of linear equations.

Condition 1: Intersecting Lines
If   a 1 a 2 ≠  b 1  b 2   ,  then the pair of linear equations a 1x + b 1y + c 1 = 0, a 2x + b 2y + c 2 = 0 has a unique solution.

Condition 2: Coincident Lines
If   a 1 a 2 =  b 1  b 2 =  c 1  c 2   , then the pair of linear equations a 1x + b 1y + c 1 = 0, a 2x + b 2y + c 2 = 0 has infinite solutions.

A pair of linear equations, which has a unique or infinite solutions are said to be a consistent pair of linear equations.

Condition 3: Parallel Lines
If   a 1 a 2 =  b 1  b 2 ≠  c 1  c 2     , then a pair of linear equations  a 1x + b 1y + c 1 = 0, a 2x + b 2y + c 2 = 0 has no solution.

A pair of linear equations which has no solution is said to be an inconsistent pair of linear equations.

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