Question 173085
Is the equation supposed to be {{{5x + 4y = 4}}}. I'll assume that it is.





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



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



{{{4y=-5x+4}}} Rearrange the terms.



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



{{{y=((-5)/(4))x+(4)/(4)}}} Break up the fraction.



{{{y=-(5/4)x+1}}} Reduce.



We can see that the equation {{{y=-(5/4)x+1}}} has a slope {{{m=-5/4}}} and a y-intercept {{{b=1}}}.



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\(-1,9\right\)].



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



{{{y-9=(4/5)(x--1)}}} Plug in {{{m=4/5}}}, {{{x[1]=-1}}}, and {{{y[1]=9}}}



{{{y-9=(4/5)(x+1)}}} Rewrite {{{x--1}}} as {{{x+1}}}



{{{y-9=(4/5)x+(4/5)(1)}}} Distribute



{{{y-9=(4/5)x+4/5}}} Multiply



{{{y=(4/5)x+4/5+9}}} Add 9 to both sides. 



{{{y=(4/5)x+49/5}}} Combine like terms. note: If you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>.



So the equation of the line perpendicular to {{{5x+4y=4}}} that goes through the point *[Tex \LARGE \left\(-1,9\right\)] is {{{y=(4/5)x+49/5}}}.



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+1,(4/5)x+49/5)
circle(-1,9,0.08),
circle(-1,9,0.10),
circle(-1,9,0.12))}}}Graph of the original equation {{{y=-(5/4)x+1}}} (red) and the perpendicular line {{{y=(4/5)x+49/5}}} (green) through the point *[Tex \LARGE \left\(-1,9\right\)].