For what values of a and b is the line -2x + y = b tangent to the curve y=ax^3 when x=5? No decimal answers.

Hello, [latex]y=2x+b \\\\ y=ax^3\\\\ slope\ =\ 3ax^2\\ if \ x=5\ then\ slope=75a\ =2\\\\ a= \dfrac{2}{75}\ and \ y= \dfrac{2}{75}* x^3\\\\ if \ x=5\ then\ y=\dfrac{2}{75}* 5^3= \dfrac{10}{3} \\\\ \dfrac{10}{3}=2*5+b\\\\ b= \dfrac{-20}{3} [/latex]

the tangent of the curve is determined by getting the first derivative of teh equation of the curve and substituting with  the given data. in this case, the derivative of y=ax^3 is y' = 3a x^2. when x is equal to 5, y' = 3a (25) = 75 a x=5; y = 125 a y-y1 = m(x-x1) y-125 a = 75 a (x-5) y = 75 ax -500 -2x + y = b -2(75/2) x  + y = -500 a = 75/2 b = -500