Question 145656


{{{2x+y=7}}} Start with the given equation.



{{{y=-2*x+7}}} Solve for y.



We can see that the equation {{{y=-2*x+7}}} has a slope {{{m=-2}}} and a y-intercept {{{b=7}}}.



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=-2}}} and the coordinates of the given point *[Tex \LARGE \left\(4,4\right\)].



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



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



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



{{{y-4=(1/2)x-2}}} Multiply



{{{y=(1/2)x-2+4}}} Add 4 to both sides. 



{{{y=(1/2)x+2}}} Combine like terms. 



So the equation of the line perpendicular to {{{2x+y=7}}} that goes through the point *[Tex \LARGE \left\(4,4\right\)] is {{{y=(1/2)x+2}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-2*x+7,(1/2)x+2)
circle(4,4,0.08),
circle(4,4,0.10),
circle(4,4,0.12))}}}Graph of the original equation {{{y=-2*x+7}}} (red) and the perpendicular line {{{y=(1/2)x+2}}} (green) through the point *[Tex \LARGE \left\(4,4\right\)].