Question 706615
{{{6x-3y = 8}}}.....first write it in {{{slope-intercept}}} form

{{{6x-8 = 3y}}}

{{{6x/3-8/3 = y}}}

{{{2x-2.67 = y}}}......so, the slope is {{{m=2}}}


if {{{g(2) = 2}}}, the {{{x=2}}} and {{{y=2}}}; so, you have a point ({{{2}}},{{{2}}}) 

you also have a line {{{2x-2.67 = y}}}  and the line you are looking for is perpendicular to this line


now, we can find the linear function satisfying the given conditions


Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of {{{2}}}, you can find the perpendicular slope by this formula:

{{{m[p]=-1/m}}} where {{{m[p]}}} is the perpendicular slope; so, plug in the given slope to find the perpendicular slope

{{{highlight(m[p]=-1/2)}}}


now we know the slope of the unknown line ({{{-1/2)}}}) and a point ({{{x[1]}}},{{{y[1]}}})=({{{2}}},{{{2}}}); so,  we can find the equation by plugging in this info into the point-slope formula
 
{{{y-y[1]=m[p](x-x[1])}}}....plug in {{{m[p]=-1/2}}},{{{x[1]=2}}}, and {{{y[1]=2}}}

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

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

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

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

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

or

{{{g(x)=-(1/2)x+3}}}...........this is a line that is perpendicular to the given graph and goes through ({{{2}}},{{{2}}})


let's see it on a graph:


{{{ graph(600, 600, -10, 10, -10, 10, 2x-2.67 , -(1/2)x+3) }}}