Question 146191

{{{5x + 4y = 24}}} Start with the given equation.



{{{4y=24-5x}}} Subtract {{{5x}}} from both sides.



{{{y=(24-5x)/(4)}}} Divide both sides by {{{4}}} to isolate {{{y}}}.



{{{y=-(5/4)x+6}}} Rearrange the terms and simplify.



We can see that the equation {{{y=-(5/4)x+6}}} has a slope {{{m=-5/4}}} and a y-intercept {{{b=6}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=-5/4}}} to get {{{m=-4/5}}}. Now change the sign to get {{{m=4/5}}}. So the perpendicular slope is {{{m=4/5}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=-5/4}}} and the coordinates of the given point *[Tex \LARGE \left\(-5,4\right\)].



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



{{{y-4=(4/5)(x--5)}}} Plug in {{{m=4/5}}}, {{{x[1]=-5}}}, and {{{y[1]=4}}}



{{{y-4=(4/5)(x+5)}}} Rewrite {{{x--5}}} as {{{x+5}}}



{{{y-4=(4/5)x+(4/5)(5)}}} Distribute



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



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



{{{y=(4/5)x+8}}} Combine like terms. 



So the equation of the line perpendicular to {{{5x + 4y = 24}}} that goes through the point *[Tex \LARGE \left\(-5,4\right\)] is {{{y=(4/5)x+8}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-(5/4)x+6,(4/5)x+8)
circle(-5,4,0.08),
circle(-5,4,0.10),
circle(-5,4,0.12))}}}Graph of the original equation {{{y=-(5/4)x+6}}} (red) and the perpendicular line {{{y=(4/5)x+8}}} (green) through the point *[Tex \LARGE \left\(-5,4\right\)].