The simplest system of dependent linear equations of two variables x and y is: ax+by = c and k(ax+by) = kc.
The equations of this type could be derived from one another by a suitable multiplication or division by a constant.
Since both equations represent same equation, the coordinates of all the points on the graph are the solutions. So there are infinitely many solutions.
The condition for dependent equations a1x+b1x+c1 = 0 and a2 x+b2x+c2 = 0 is:
a2/a1 = b2/b2=c2/c1 = l a finite number.
Example: x+2y = 3 and 3x+6y = 3.
Both equations could be written like :
x+2y-3 = 0 and 3x+6y-9 = 0.
3/1= 6/2 = -9/-3 = 3.
Therefore they are dependent system of equations.
The solutions are many:
Some solutions are:
(x,y) = (1,1), (2, -1/2), (5 , -1), ( 7, -2) etc.
Dependent equations describe the same line, though they are different. The system formed by dependent equations has an infinite number of solutions.
3x + 4y = 2
6x + 8y = 4
We notice that we get the second equation multiplying by 2 the first equation.
So, both equations are equivalent and all points located on a line, they are located on the other line also.
The system has an infinite number of solutions.