Question 184600

{{{6x-4y=16}}} Start with the given equation.



{{{-4y=16-6x}}} Subtract {{{6x}}} from both sides.



{{{-4y=-6x+16}}} Rearrange the terms.



{{{y=(-6x+16)/(-4)}}} Divide both sides by {{{-4}}} to isolate y.



{{{y=((-6)/(-4))x+(16)/(-4)}}} Break up the fraction.



{{{y=(3/2)x-4}}} Reduce.



We can see that the equation {{{y=(3/2)x-4}}} has a slope {{{m=3/2}}} and a y-intercept {{{b=-4}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=3/2}}} to get {{{m=2/3}}}. Now change the sign to get {{{m=-2/3}}}. So the perpendicular slope is {{{m=-2/3}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=3/2}}} and the coordinates of the given point *[Tex \LARGE \left\(-4,6\right\)].



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



{{{y-6=(-2/3)(x--4)}}} Plug in {{{m=-2/3}}}, {{{x[1]=-4}}}, and {{{y[1]=6}}}



{{{y-6=(-2/3)(x+4)}}} Rewrite {{{x--4}}} as {{{x+4}}}



{{{y-6=(-2/3)x+(-2/3)(4)}}} Distribute



{{{y-6=(-2/3)x-8/3}}} Multiply



{{{y=(-2/3)x-8/3+6}}} Add 6 to both sides. 



{{{y=(-2/3)x+10/3}}} 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 {{{6x-4y=16}}} that goes through the point *[Tex \LARGE \left\(-4,6\right\)] is {{{y=(-2/3)x+10/3}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,(3/2)x-4,(-2/3)x+10/3)
circle(-4,6,0.08),
circle(-4,6,0.10),
circle(-4,6,0.12))}}}Graph of the original equation {{{y=(3/2)x-4}}} (red) and the perpendicular line {{{y=(-2/3)x+10/3}}} (green) through the point *[Tex \LARGE \left\(-4,6\right\)].