#### Answer

$a.$ The slope is $m_{tan} = -7$.
$b.$ The equation is $y=-7x$.

#### Work Step by Step

$a.$ Using the formula from definition (2) with $a=-1$ and $f(a)=7$ (coordinates of the point $P(-1,7)$) we have
$$m_{tan}=\lim_{h\to0}\frac{f(-1+h)-f(-1)}{h}=\lim_{h\to0}\frac{-7(-1+h)-7}{h}=\lim_{h\to0}\frac{7-7h-7}{h}=\lim_{h\to0}\frac{-7h}{h}=\lim_{h\to0}-7h =-7.$$
$b.$ Using the formula $y-f(a)=m_{tan}(x-a)$ with the same values for $a$ and $f(a)$ as in part $a$ and the calculated value $m_{tan} = -7$ we get
$$y-7=-7(x-(-1))\Rightarrow y-7=-7x-7$$ which gives
$$y=-7x.$$