There are TWO solutions to this problem, for there are two ways a
line can have an angle of 60 degrees with the line x=4.

Here is one solution:

We are to find the equation of the black line:



We need the slope of the black line.  The slope is 
We have the RISE which is BC or 7.  We need the RUN.
Since the triangle ABC is a 30-60-90 right triangle, 

AC = 2(BC) = 2(7) = 14

AB²+BC² = AC²
 AB²+7² = 14²
 AB²+49 = 196
    AB² = 147
     AB = √147
     AB = √49*3
     AB = 7√3
Now we have the RUN.

So the slope is 

The black line goes through (4,7), so we use the point-slope form:

y - y1 = m(x - x1)

where (x1,y1) = (4,-7) and m = 

y - (-7) = sqrt(3)/3}}}(x - 4)

   y + 7 = sqrt(3)/3}}}x - 4sqrt(3)/3}}}

       y = sqrt(3)/3}}}x - 4sqrt(3)/3}}} - 7

That's one solution.  The other solution is done this way:

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The slope of that black line will be the negative of the slope of the
black line in the first solution.  That is m =   I'll 
let you do this second solution.

Edwin