Given, the equation of lines are:

2x - 3y + 4 = 0 ...........1

3x + 4y - 5 = 0 ...........2

6x - 7y + 8 = 0 ...........3

The person is standing at the junction of the paths represented by lines 1 and 2

After solving eqaution 1 and 2, we get

x = -1/17 and y = 22/17

So, the person is standing at point (-1/17, 22/17)

Again, the person can reach the path 3 in the least time if he walks along the perpendicular line to 3 from the point (-1/17, 2/17)

Now, slope of the line 3  = 6/7

Slope of the line perpendicular to the line 3 = -1/(6/7) = -7/6

Now, the equation of the line passing through the point (-1/17, 22/17) and having slope  -7/6 is

      y - 22/17 = (-7/6)*(x + 1/17)

=> (17y - 22)/17 = (-7/6)*(17x + 1)/17

=> (17y - 22) = (-7/6)*(17x + 1)

=> 6(17y - 22) = (-7)*(17x + 1) 

=> 102y - 132 = -119x - 7

=> 119x + 102y = 125

Hense, the path that the person should follow is 119x + 102y = 125