Question 243550


{{{4x-3y=7}}} Start with the given equation.



{{{-3y=7-4x}}} Subtract {{{4x}}} from both sides.



{{{-3y=-4x+7}}} Rearrange the terms.



{{{y=(-4x+7)/(-3)}}} Divide both sides by {{{-3}}} to isolate y.



{{{y=((-4)/(-3))x+(7)/(-3)}}} Break up the fraction.



{{{y=(4/3)x-7/3}}} Reduce.



We can see that the equation {{{y=(4/3)x-7/3}}} has a slope {{{m=4/3}}} and a y-intercept {{{b=-7/3}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=4/3}}} to get {{{m=3/4}}}. Now change the sign to get {{{m=-3/4}}}. So the perpendicular slope is {{{m=-3/4}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=-3/4}}} and the coordinates of the given point *[Tex \LARGE \left\(5,-2\right\)].



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



{{{y--2=(-3/4)(x-5)}}} Plug in {{{m=-3/4}}}, {{{x[1]=5}}}, and {{{y[1]=-2}}}



{{{y+2=(-3/4)(x-5)}}} Rewrite {{{y--2}}} as {{{y+2}}}



{{{y+2=(-3/4)x+(-3/4)(-5)}}} Distribute



{{{y+2=(-3/4)x+15/4}}} Multiply



{{{y=(-3/4)x+15/4-2}}} Subtract 2 from both sides. 



{{{y=(-3/4)x+7/4}}} 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 {{{4x-3y=7}}} that goes through the point *[Tex \LARGE \left\(5,-2\right\)] is {{{y=(-3/4)x+7/4}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,(4/3)x-7/3,(-3/4)x+7/4)
circle(5,-2,0.08),
circle(5,-2,0.10),
circle(5,-2,0.12))}}}


Graph of the original equation {{{y=(4/3)x-7/3}}} (red) and the perpendicular line {{{y=(-3/4)x+7/4}}} (green) through the point *[Tex \LARGE \left\(5,-2\right\)].