What value of b will cause the system to have an infinite number of solutions?
y = 6x + b
-3x+ 1/2 y = -3

b must be equal to -6  for infinitely many solutions for system of equations and

Solution:

Need to calculate value of b so that given system of equations have an infinite number of solutions

Let us bring the equations in same form for sake of simplicity in comparison

Now we have two equations  

Let us first see what is requirement for system of equations have an infinite number of solutions

If   and are two equation  

then the given system of equation has no infinitely many solutions.

In our case,

 As for infinitely many solutions

Hence b must be equal to -6 for infinitely many solutions for system of equations and  

i hope this is what you were looking for.

see attached graph