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.
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.
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;
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