Aptitude - Basic Equations

Linear equations in two variables

An equations of the form ax + by +c= 0, where a, b, c ⊂R and a≠0 , b≠0 and x ,y are variables, is called a linear equation in two variables.

Solution: Any pair of values of x and y which satisfy the equation ax + by + c =0, is called its solution.

Consistent and inconsistent system of linear Equations

A system consisting of two simultaneous linear equations is said to be:

• Consistent, if it has at least one solution.

• Inconsistent, if it has no solution.

Conditions for Solvability

The system of equation a₁x+ b₁y+c₁=0, a₂x + b₂y+ c₂= 0 has

• A unique solution , if a₁/a₂ ≠ b₁/b₂ ;

• An infinite number of solutions, if a₁/a₂ = b₁/b₂= c₁/c₂;

• No solution , if a₁/a₂ = b₁/b₂≠ c₁/c₂;

Homogeneous system of equations

The system of equations a₁x+ b₁y= 0; a₂x+ b₂y = 0 has

• Only solution x= 0 , y= 0 when a₁/a₂ ≠ b₁/b₂;

• An infinite number of solutions when a₁/a₂ = b₁/b₂

Solved Examples