Question 199963

{{{9x+2y=18}}} Start with the given equation.



{{{2y=18-9x}}} Subtract {{{9x}}} from both sides.



{{{2y=-9x+18}}} Rearrange the terms.



{{{y=(-9x+18)/(2)}}} Divide both sides by {{{2}}} to isolate y.



{{{y=((-9)/(2))x+(18)/(2)}}} Break up the fraction.



{{{y=-(9/2)x+9}}} Reduce.



We can see that the equation {{{y=-(9/2)x+9}}} has a slope {{{m=-9/2}}} and a y-intercept {{{b=9}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=-9/2}}} to get {{{m=-2/9}}}. Now change the sign to get {{{m=2/9}}}. So the perpendicular slope is {{{m=2/9}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=-9/2}}} and the coordinates of the given point *[Tex \LARGE \left\(1,4\right\)].



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



{{{y-4=(2/9)(x-1)}}} Plug in {{{m=2/9}}}, {{{x[1]=1}}}, and {{{y[1]=4}}}



{{{y-4=(2/9)x+(2/9)(-1)}}} Distribute



{{{y-4=(2/9)x-2/9}}} Multiply



{{{y=(2/9)x-2/9+4}}} Add 4 to both sides. 



{{{y=(2/9)x+34/9}}} 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 {{{9x+2y=18}}} that goes through the point *[Tex \LARGE \left\(1,4\right\)] is {{{y=(2/9)x+34/9}}}.



If you want the equation in standard form, then the equation is {{{2x-9y=34}}}



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-(9/2)x+9,(2/9)x+34/9)
circle(1,4,0.08),
circle(1,4,0.10),
circle(1,4,0.12))}}} Graph of the original equation {{{y=-(9/2)x+9}}} (red) and the perpendicular line {{{y=(2/9)x+34/9}}} (green) through the point *[Tex \LARGE \left\(1,4\right\)].