Equation of a line that has to be tangent

[QUITE RIGHT]

How would find the equation of a line that has to be tangent to a curve and parallel to another line (i know slope has to be equal)

(you are given the equation of the line and the curve)
x^3
3x-y-6

 

[Mentallic]

Ok so you know the gradients have to be the same, so wherever the gradient is 3 on the curve (finding where this happens will be done by using the derivative) will be used to find the equation of the new line.
Taking the derivative of the curve will give you $$y'=3x^2$$

and since this has to be equal to the gradient of the line which is 3, just solve $$3x^2=3$$.
Notice you'll have 2 answers, and that is how it should be!

And use the point-gradient form for the equation of a line:
$$y-y_1=m(x-x_1)$$