Question 181883


We can see that the equation {{{y=-4*x+8}}} has a slope {{{m=-4}}} and a y-intercept {{{b=8}}}.



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=-4}}} and the coordinates of the given point *[Tex \LARGE \left\(2,-3\right\)].



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



{{{y--3=(1/4)(x-2)}}} Plug in {{{m=1/4}}}, {{{x[1]=2}}}, and {{{y[1]=-3}}}



{{{y+3=(1/4)(x-2)}}} Rewrite {{{y--3}}} as {{{y+3}}}



{{{4(y+3)=x-2}}} Multiply both sides by 4.



{{{4y+12=x-2}}} Distribute



{{{4y=x-2-12}}} Subtract 12 from both sides.



{{{4y-x=-2-12}}} Subtract "x" from both sides.



{{{4y-x=-14}}} Combine like terms.



{{{-x+4y=-14}}} Rearrange the terms.



{{{x-4y=14}}} Multiply EVERY term by -1 to make the "x" coefficient positive.



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Answer: 


So the equation of the line that is perpendicular to {{{y=-4*x+8}}} and goes through the point (2,-3) in standard form is {{{x-4y=14}}}




Here's the graph of the two lines to verify the answer:



{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-4*x+8,(14-x)/(-4))
circle(2,-3,0.08),
circle(2,-3,0.10),
circle(2,-3,0.12))}}}Graph of the original equation {{{y=-4*x+8}}} (red) and the perpendicular line {{{x-4y=14}}} (green) through the point *[Tex \LARGE \left\(2,-3\right\)].