Question 253690


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



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



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



We can see that the equation {{{y=-4*x+7}}} has a slope {{{m=-4}}} and a y-intercept {{{b=7}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=-4}}} to get {{{m=-1/4}}}. Now change the sign to get {{{m=1/4}}}. So the perpendicular slope is {{{m=1/4}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=1/4}}} and the coordinates of the given point *[Tex \LARGE \left\(-4,-3\right\)].



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



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



{{{y--3=(1/4)(x+4)}}} Rewrite {{{x--4}}} as {{{x+4}}}



{{{y+3=(1/4)(x+4)}}} Rewrite {{{y--3}}} as {{{y+3}}}



{{{y+3=(1/4)x+(1/4)(4)}}} Distribute



{{{y+3=(1/4)x+1}}} Multiply



{{{y=(1/4)x+1-3}}} Subtract 3 from both sides. 



{{{y=(1/4)x-2}}} Combine like terms. 



So the equation of the line perpendicular to {{{4x+y=7}}} that goes through the point *[Tex \LARGE \left\(-4,-3\right\)] is {{{y=(1/4)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,-4*x+7,(1/4)x-2)
circle(-4,-3,0.08),
circle(-4,-3,0.10),
circle(-4,-3,0.12))}}}


Graph of the original equation {{{y=-4*x+7}}} (red) and the perpendicular line {{{y=(1/4)x-2}}} (green) through the point *[Tex \LARGE \left\(-4,-3\right\)].