**Important Concepts and Formulas**

1. Two linear equations in the same two
variables are called a pair of linear equations in two variables. The most
general form of a pair of linear equations is

*a*= 0

_{1}x + b_{1}y + c_{1}*a*= 0

_{2}x + b_{2}y + c_{2}
where

*a*are real numbers such that_{1, }a_{2, }b_{1, }b_{2, }c_{1, }c_{2 }*a*+_{1}^{2}_{ }*b*≠ 0 and_{1}^{2}_{ }*a*+_{2}^{2}_{ }*b*≠ 0._{2}^{2}_{ }
2. A pair of linear equations in two
variables can be represented and solved by the following methods:

i.
Graphical
method

ii.
Algebraic
method

3.

**Graphical method:**The graph of a pair of linear equations in two variables is represented by two lines.
i.
If
the lines intersect at a point, then that point gives the unique solution of
the two equations. In this case, the pair of equations is

**consistent**.
ii.
If
the lines coincide, then there are infinitely many solutions, i.e., each point
on the line being a solution. In this case, the pair the equations is

**dependent**(**consistent**).
iii.
If
the lines are parallel, then the pair of equations has no solution. In this
case, the pair of equations is

**inconsistent**.
4.

**Algebraic method:**We use the following methods to find the solution of a pair of linear equations:
iii.
Cross-multiplication

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5. If a pair of linear equations is
given by

*a*= 0 and_{1}x + b_{1}y + c_{1}*a*= 0, then the following situations can arise:_{2}x + b_{2}y + c_{2}
i.
If a

_{1}/ a_{2 }≠ b_{1}/ b_{2}, then both the equations have a unique solution and they are consistent.
ii.
If a

_{1}/ a_{2 }= b_{1}/ b_{2 }= c_{1}/c_{2}, then both the equations have infinitely many solutions and they are dependent and consistent.
iii.
If a

_{1}/ a_{2 }= b_{1}/ b_{2 }≠ c_{1}/c_{2}, then both the equations have no solution and the equations are said to be inconsistent.