Either your maths teacher has got it wrong(!) or you've copied the answer incorrectly. (Actually, you seem to have copied the question incorrectly as well. You appear to be seeking an equation, not a gradient, as the answer to your problem).
The graph of y = mx +c always has a gradient equal to m (and a y-intercept equal to c). So y = 3x + 5 must have a gradient of 3.
Since the required line is parallel to y = 3x + 5, it must also have a gradient of 3.
i.e. the required line is of the form y = 3x + c
All we need to do is to find the value of c. We know that the line passes through (4,3) so we know that when x = 4, y =3. So let's put those values into y = 3x + c:
We get 3 = (3x4) +c
<=> 3 = 12 + c
<=> c = -9
Putting that into y = 3x + c gives us
y = 3x - 9
Chris
(Maths graduate with 15 years teaching experience - so I hope I've got it right!)