When we change the equations to slope-intercept form...

    2x - y = 1 becomes y = 2x - 1
            slope = 2
            y-intercept = -1

    6x - 3y = 3 becomes y = 2x - 1
            slope = 2
            y-intercept = -1

When we graph this it appears like that below:

Since the two lines coincide there is an infinite number of solutions to this system of equations.  We can also tell this by looking at the slopes and y-intercepts of the lines.  Although the slopes are the same, just like parallel lines, so are the y-intercepts which means that the two equations are the same line.  You can choose any point on the line and it should satisfy both original equations.