SOLUTION: find the distance (rounded to the nearest thousandth) from the point (-5,-2) to the line 3x+y=8. The distance from a point to a line is the perpendicular distance.

Algebra ->  Length-and-distance -> SOLUTION: find the distance (rounded to the nearest thousandth) from the point (-5,-2) to the line 3x+y=8. The distance from a point to a line is the perpendicular distance.      Log On


   

Question 121645: find the distance (rounded to the nearest thousandth) from the point (-5,-2) to the line 3x+y=8. The distance from a point to a line is the perpendicular distance.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

First convert the standard equation 3x%2B1y=8 into slope intercept form

Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)
Convert from standard form (Ax+By = C) to slope-intercept form (y = mx+b)


3x%2B1y=8 Start with the given equation


3x%2B1y-3x=8-3x Subtract 3x from both sides


1y=-3x%2B8 Simplify


The original equation 3x%2B1y=8 (standard form) is equivalent to y+=+-3x%2B8 (slope-intercept form)


The equation y+=+-3x%2B8 is in the form y=mx%2Bb where m=-3 is the slope and b=8 is the y intercept.







Now let's find the equation of the line that is perpendicular to y=-3x%2B8 which goes through (-5,-2)

Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line


Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of -3, you can find the perpendicular slope by this formula:

m%5Bp%5D=-1%2Fm where m%5Bp%5D is the perpendicular slope


m%5Bp%5D=-1%2F%28-3%2F1%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%281%2F-3%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=1%2F3 Multiply the fractions.


So the perpendicular slope is 1%2F3



So now we know the slope of the unknown line is 1%2F3 (its the negative reciprocal of -3 from the line y=-3%2Ax%2B8). Also since the unknown line goes through (-5,-2), we can find the equation by plugging in this info into the point-slope formula

Point-Slope Formula:

y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope and (x%5B1%5D,y%5B1%5D) is the given point



y%2B2=%281%2F3%29%2A%28x%2B5%29 Plug in m=1%2F3, x%5B1%5D=-5, and y%5B1%5D=-2



y%2B2=%281%2F3%29%2Ax-%281%2F3%29%28-5%29 Distribute 1%2F3



y%2B2=%281%2F3%29%2Ax%2B5%2F3 Multiply



y=%281%2F3%29%2Ax%2B5%2F3-2Subtract -2 from both sides to isolate y

y=%281%2F3%29%2Ax%2B5%2F3-6%2F3 Make into equivalent fractions with equal denominators



y=%281%2F3%29%2Ax-1%2F3 Combine the fractions



y=%281%2F3%29%2Ax-1%2F3 Reduce any fractions

So the equation of the line that is perpendicular to y=-3%2Ax%2B8 and goes through (-5,-2) is y=%281%2F3%29%2Ax-1%2F3


So here are the graphs of the equations y=-3%2Ax%2B8 and y=%281%2F3%29%2Ax-1%2F3




graph of the given equation y=-3%2Ax%2B8 (red) and graph of the line y=%281%2F3%29%2Ax-1%2F3(green) that is perpendicular to the given graph and goes through (-5,-2)






So by using substitution, elimination, or graphing, we find that the two lines intersect at the point (2.5,0.5)

Now let's use the distance formula to find the distance between the two points (-5,-2) and (2.5,0.5)


Start with the given distance formula
d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 where is the first point and is the second point

d=sqrt%28%28-5-2.5%29%5E2%2B%28-2-0.5%29%5E2%29 Plug in x%5B1%5D=-5, x%5B2%5D=2.5, y%5B1%5D=-2, y%5B2%5D=0.5

d=sqrt%28%28-7.5%29%5E2%2B%28-2.5%29%5E2%29 Evaluate -5-2.5 to get -7.5. Evaluate -2-0.5 to get -2.5.

d=sqrt%2856.25%2B6.25%29 Square each value

d=sqrt%2862.5%29 Add

So the distance approximates to

d=7.90569415042095

which rounds to
d=7.906



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Answer:
So the distance between the point (-5,-2) and the line 3x+y=8 is approximately 7.906 units