Question 145614


{{{8x+5y=3}}} Start with the given equation.



{{{y=-(8/5)x+3/5}}} Solve for y.



We can see that the equation {{{y=-(8/5)x+3/5}}} has a slope {{{m=-8/5}}} and a y-intercept {{{b=3/5}}}.



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\(0,5\right\)].



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



{{{y-5=(5/8)(x-0)}}} Plug in {{{m=5/8}}}, {{{x[1]=0}}}, and {{{y[1]=5}}}



{{{y-5=(5/8)x+(5/8)(-0)}}} Distribute



{{{y-5=(5/8)x+0}}} Multiply



{{{y=(5/8)x+0+5}}} Add 5 to both sides. 



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



So the equation of the line perpendicular to {{{8x+5y=3}}} that goes through the point *[Tex \LARGE \left\(0,5\right\)] is {{{y=(5/8)x+5}}}.



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+3/5,(5/8)x+5)
circle(0,5,0.08),
circle(0,5,0.10),
circle(0,5,0.12))}}}Graph of the original equation {{{y=-(8/5)x+3/5}}} (red) and the perpendicular line {{{y=(5/8)x+5}}} (green) through the point *[Tex \LARGE \left\(0,5\right\)].