- For any linear equation, each solution (x, y) corresponds to a point on the line. General form is given by ax + by + c = 0.
- The graph of a linear equation is a straight line.
- 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: a1x + b1y + c1 = 0; a2x + b2y + c2 = 0
where a1, a2, b1, b2, c1 and c2 are real numbers, such that a12 + b12 ≠ 0, a22 + b22 ≠ 0.
- A pair of values of variables ‘x‘ and ‘y’ which satisfy both the equations in the given system of equations is said to be a solution of the simultaneous pair of linear equations.
- A pair of linear equations in two variables can be represented and solved, by
(i) Graphical method
(ii) Algebraic method
(i) Graphical method. The graph of a pair of linear equations in two variables is presented by two lines.
(ii) Algebraic methods. Following are the methods for finding the solutions(s) of a pair of linear equations:
- Substitution method
- Elimination method
- Cross-multiplication method.
- There are several situations which can be mathematically represented by two equations that are not linear to start with. But we allow them so that they are reduced to a pair of linear equations.
- Consistent system. A system of linear equations is said to be consistent if it has at least one solution.
- Inconsistent system. A system of linear equations is said to be inconsistent if it has no solution.
CONDITIONS FOR CONSISTENCY
Let the two equations be:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
|Relationship between coeff. or the pair of equations||Graph||Number of Solutions||Consistency of System|
|Intersecting lines||Unique solution||Consistent|
|Parallel lines||No solution||Inconsistent|
|Co-incident lines||Infinite solutions||Consistent|