Question 323861


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



Now to find the slope of the perpendicular line, simply flip the slope {{{m=2}}} to get {{{m=1/2}}}. Now change the sign to get {{{m=-1/2}}}. So the perpendicular slope is {{{m=-1/2}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=-1/2}}} and the coordinates of the given point *[Tex \LARGE \left\(1,3\right\)].



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



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



{{{y-3=(-1/2)x+(-1/2)(-1)}}} Distribute



{{{y-3=(-1/2)x+1/2}}} Multiply



{{{y=(-1/2)x+1/2+3}}} Add 3 to both sides. 



{{{y=(-1/2)x+7/2}}} 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 {{{y=2x-3}}} that goes through the point *[Tex \LARGE \left\(1,3\right\)] is {{{y=(-1/2)x+7/2}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,2x-3,(-1/2)x+7/2)
circle(1,3,0.08),
circle(1,3,0.10),
circle(1,3,0.12))}}}


Graph of the original equation {{{y=2x-3}}} (red) and the perpendicular line {{{y=(-1/2)x+7/2}}} (green) through the point *[Tex \LARGE \left\(1,3\right\)].