Question

What is the x intercept that is perpendicular to y=4/3x+1 and passes through (-4,9)

Answer 1

Well, we know the line is perpendicular to that one above.... what is the slope of that one anyway?  well, notice, the equation is already in slope-intercept form   $\bf y=\stackrel{slope}{\cfrac{4}{3}}x+1$.

so, we're looking for the equation of a line perpendicular to that one, now, since that one has a slope of 4/3, a perpendicular line will have a negative reciprocal slope to that one, 

$\bf \textit{perpendicular, negative-reciprocal slope for}\quad \cfrac{4}{3}\\\\ negative\implies -\cfrac{4}{ 3}\qquad reciprocal\implies - \cfrac{ 3}{4}$

so, what is the equation of a line whose slope is -3/4 and runs through -4,9?

$\bf \begin{array}{ccccccccc} &&x_1&&y_1\\ &&(~ -4 &,& 9~) \end{array} \\\\\\ % slope = m slope = m\implies -\cfrac{3}{4} \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-9=-\cfrac{3}{4}[x-(-4)] \\\\\\ y-9=-\cfrac{3}{4}(x+4)$

now, the x-intercept for any function is found by zeroing out the "y" and solving for "x", thus

$\bf y-9=-\cfrac{3}{4}(x+4)\implies 0-9=-\cfrac{3}{4}(x+4)\implies -9=-\cfrac{3x}{4}-3 \\\\\\ -6=-\cfrac{3x}{4}\implies -24=-3x\implies \cfrac{-24}{-3}=x\implies 8=x$

x = 8, y = 0           ( 8 , 0 )