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

Solved Examples
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