This is impossible. Both the slopes and the y-intercepts must be the same in
both equations in order for two equations to have an infinite number of
solutions.   

No matter what value k takes on, the slope of 5x + 7y = k is the same. To show
that they do not have the same slope, we put them both in slope-intercept
form:  "y = mx + b":

5x + 7y = k
     7y = -5x + k
    
    

Comparing with y = mx + b the slope is  and the y-intercept is
the point 

For the other equation:

10x + 19y = 3
      19y = -10x + 3
    
    
 
Comparing with y = mx + b the slope is  and the y-intercept is
the point 

They do not have the same slope, so no matter what value of k we use,
there will always be exactly one solution.

Edwin