Class-X Linear Equation in Two variables

Introduction-Linear Equation in Two variables

If two linear equations have the two same variables, they are called a pair of linear equations in two variables. Following is the most general form of linear equations:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Here, a1, a2, b1, b2, c1 and c2 are real numbers such that;

A pair of linear equations can be represented and solved by the following methods:

• Graphical method
• Algebraic method

Graphical Method:

For a given pair of linear equations in two variables, the graph is represented by two lines.

• If the lines intersect at a point, that point gives the unique solution for the two equations. If there is a unique solution of the given pair of equations, the equations are called consistent.
• If the lines coincide, there are indefinitely many solutions for the pair of linear equations. In this case, each point on the line is a solution. If there are infinitely many solutions of the given pair of linear equations, the equations are called dependent (consistent).
• If the lines are parallel, there is no solution for the pair of linear equations. If there is no solution of the given pair of linear equations, the equations are called inconsistent.

Algebraic Method:

There are following methods for finding the solutions of the pair of linear equations:

• Substitution method
• Elimination method
• Cross-multiplication method