CONCEPT OF PAIR of linear eqvation in two variable.

The concept of a pair of linear equations refers to a mathematical system consisting of two linear equations involving two variables. These equations represent straight lines in a coordinate plane, and the solution to the system is the point(s) where these two lines intersect (if any).
General Form of a Pair of Linear Equations
The general form of a linear equation in two variables is:
a1x+b1y+c1=0a_1x + b_1y + c_1 = 0a1x+b1y+c1=0
a2x+b2y+c2=0a_2x + b_2y + c_2 = 0a2x+b2y+c2=0
Here:
• xxx and yyy are the variables.
• a1,b1,c1,a2,b2,c2a_1, b_1, c_1, a_2, b_2, c_2a1,b1,c1,a2,b2,c2 are constants.
Graphical Representation
• Each equation represents a straight line in the xyxyxy-plane.
• Solving the pair of equations means finding the point(s) of intersection of these lines.
Types of Solutions
The solution depends on the relationship between the two lines:
1. Unique Solution: The two lines intersect at exactly one point.
(Lines are consistent and independent.)
2. No Solution: The two lines are parallel and never meet.
(Lines are inconsistent.)
3. Infinite Solutions: The two lines overlap completely, meaning they are coincident.
(Lines are consistent and dependent.)
Methods to Solve a Pair of Linear Equations
1. Graphical Method: Plot both equations on a graph and find their intersection.
2. Algebraic Methods:
o Substitution Method: Solve one equation for one variable and substitute it into the other equation.
o Elimination Method: Add or subtract the equations to eliminate one variable, then solve for the other.
o Cross-Multiplication Method: Use a formulaic approach for solving equations in the form a1x+b1y+c1=0a_1x + b_1y + c_1 = 0a1x+b1y+c1=0 and a2x+b2y+c2=0a_2x + b_2y + c_2 = 0a2x+b2y+c2=0.
Applications
• Real-life problems like finding the point of intersection between roads or solving economic problems involving costs and profits.
• Used in physics, engineering, economics, and optimization problems.
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