Question 221112
Method #1 


{{{8x+5y=6}}} Start with the given equation.


ANY equation perpendicular to the equation above will be in the form {{{5x-8y=c}}} (note: just swap the coefficients and negate the second coefficient). So just plug in (8,2) to find 'c':


{{{5(8)-8(-2)=40+16=56}}}. So {{{c=56}}} which means that the perpendicular equation is {{{5x-8y=56}}} which in slope-intercept form is {{{y=(5/8)x-7}}}



So you were thinking of something similar to this (actually I think you were trying to use a rule for parallel equations).



If the method above made no sense, then check out this alternative...


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Method #2



{{{8x+5y=6}}} Start with the given equation.



{{{5y=6-8x}}} Subtract {{{8x}}} from both sides.



{{{5y=-8x+6}}} Rearrange the terms.



{{{y=(-8x+6)/(5)}}} Divide both sides by {{{5}}} to isolate y.



{{{y=((-8)/(5))x+(6)/(5)}}} Break up the fraction.



{{{y=-(8/5)x+6/5}}} Reduce.



We can see that the equation {{{y=-(8/5)x+6/5}}} has a slope {{{m=-8/5}}} and a y-intercept {{{b=6/5}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=-8/5}}} to get {{{m=-5/8}}}. Now change the sign to get {{{m=5/8}}}. So the perpendicular slope is {{{m=5/8}}}.



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



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



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



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



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



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



{{{y=(5/8)x-5-2}}} Subtract 2 from both sides. 



{{{y=(5/8)x-7}}} Combine like terms. 



So the equation of the line perpendicular to {{{8x+5y=6}}} that goes through the point *[Tex \LARGE \left\(8,-2\right\)] is {{{y=(5/8)x-7}}}. This equation is {{{5x-8y=56}}} in standard form.



So the answer is either {{{y=(5/8)x-7}}} or {{{5x-8y=56}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-(8/5)x+6/5,(5/8)x-7)
circle(8,-2,0.08),
circle(8,-2,0.10),
circle(8,-2,0.12))}}} 


Graph of the original equation {{{y=-(8/5)x+6/5}}} (red) and the perpendicular line {{{y=(5/8)x-7}}} (green) through the point *[Tex \LARGE \left\(8,-2\right\)].