The thick red lines enclose the given constrained region. The blue line is of the form x+y=c where c is a constant. c has a minimum value when the line is as far over to the left as possible to remain in the enclosed region.
The line 2x-3y=-1 is also 3y-2x=1, and has a y-intercept at y=⅓. The blue line must pass through this intercept to be in its leftmost position. So x+y=⅓ is the equation and c=⅓ is the minimum value.