Question 227558

We can see that the equation {{{y=(5/2)x-17}}} has a slope {{{m=5/2}}} and a y-intercept {{{b=-17}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=5/2}}} to get {{{m=2/5}}}. Now change the sign to get {{{m=-2/5}}}. So the perpendicular slope is {{{m=-2/5}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=-2/5}}} and the coordinates of the given point *[Tex \LARGE \left\(-10,8\right\)].



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-8=(-2/5)(x--10)}}} Plug in {{{m=-2/5}}}, {{{x[1]=-10}}}, and {{{y[1]=8}}}



{{{y-8=(-2/5)(x+10)}}} Rewrite {{{x--10}}} as {{{x+10}}}



{{{y-8=(-2/5)x+(-2/5)(10)}}} Distribute



{{{y-8=(-2/5)x-4}}} Multiply



{{{y=(-2/5)x-4+8}}} Add 8 to both sides. 



{{{y=(-2/5)x+4}}} Combine like terms. 



So the equation of the line perpendicular to {{{y=(5/2)x-17}}} that goes through the point *[Tex \LARGE \left\(-10,8\right\)] is {{{y=(-2/5)x+4}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -12, 10, -10, 10,
graph(500, 500, -12, 10, -10, 10,(5/2)x-17,(-2/5)x+4)
circle(-10,8,0.08),
circle(-10,8,0.10),
circle(-10,8,0.12))}}}


Graph of the original equation {{{y=(5/2)x-17}}} (red) and the perpendicular line {{{y=(-2/5)x+4}}} (green) through the point *[Tex \LARGE \left\(-10,8\right\)].