Graphical Method of Solution
* The general form of a pair of a linear equations in two variables x and y are:
a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} =0
a_{1}, a_{2}, b_{1} , b_{2} , c_{1} , c_{2} are all real numbers and a_{1}² + b_{1}² ≠ 0 and a_{2}² + b_{2}² ≠ 0
Solving a pair of linear equation graphically:
* Frame the pair of linear equations based on the problem
* Find ordered pair (x, y) satisfying the equations
* Plot the pints obtained on the graph to find the solutions
Consistent solution: a pair of linear equations having unique or infinite solutions.
Unique solutions: If \frac{ a_{1} }{ a_{2} } ≠ \frac{ b_{1} }{ b_{2} },
then a pair of linear equations has a unique solution and they intersect at only one point.1
Infinite solution: If \frac{ a_{1} }{ a_{2} } = \frac{ b_{1} }{ b_{2} } = \frac{ c_{1} }{ c_{2} } ,
then the pairs of linear equation has infinite solutions and they are coincident with each other.
Inconsistent solution: a pair of linear equations which have no solutions.
No solutions: if \frac{ a_{1} }{ a_{2} } = \frac{ b_{1} }{ b_{2} } ≠ \frac{ c_{1} }{ c_{2} } ,
then the pair of linear equations has no solutions and they do not intersect with each other.
Algebraic Methods of Solving
A pair of linear equation can be solved using any of the following methods
* Substitution method
* Elimination method
* Cross-multiplication method
We can solve for any pair of non-linear equations, if such equation can be represented as a pair of linear equations