We put the line

4x + y = k

in slope-intercept form y = mx + b

4x + y = k
     y = -4x + k

So it has slope m = -4.

That means that any line perpendicular to it has
the slope m = , the "negative reciprocal"
of the -4, i.e., "invert and change the sign".

Now we can find the equation of the line through
the origin (0,0) with slope .

We use the point-slope formula:
y - y1 = m(x - x1)
where (x1,y1) = (0,0)
y - 0 = (x-0)

which simplifies to

y = x

That will be easier to work with if we 
clear the fraction by multiplying both
sides by 4

4y = x 

Now we have this system of equations:

     y = -4x + k
    4y = x

which we are told it has solution (t,t+1),
their given point of intersection.

So we substitute t for x and t+1 for y:

     t+1 = -4t + k
    4(t+1) = t

which simplify to:

    5t-k = -1 
    4t+4 = t

and the second one simplifies further and we have:

    5t-k = -1
      3t = -4

We solve the 2nd equation for t

       t = 

That's all you wanted.  However you can continue and
find k too

Substituting  for t the first equation, we have

    5t-k = -1
    5()-k = -1
    -k = -1
    -20-3k = -3
       -3k = 17
         k = 

But all you wanted was  t = 

Edwin