I dont understand this

Question

Answer 1

solve the equation for y

y =-2x/5 +1

find the perpendicular line to the given line through the given point

y =5x/2 +b

-1 =-20/2 +b

b = -1+10 = 9

perpendicular line is y=5x/2 +9

find where the two lines intersect

set the y's equal

5x/2 +9 = -2x/5 +1

5x/2 +2x/5 = 1-9

29x/5 = -8

x = -8(5)/29 =-40/29

y = 5(-40/29)/2 + 9 = -100/29+9 = (-100+261)/29=161/29

find the distance from the intersection point to the original point

from (-40/29, 161/29) to (-4,-1)

(-40/29, 161/29) is the point on the line that's the shortest distance to (-4,-1)

= about (-1.34, 5.55)

use the distance formula, a version of the Pythagorean Theorem

distance square = hypotenuse squared = d^2 = a^2 +b^2= sum of squares of the sides

= (-40/29+4)^2 + (161/29+1)

= (-40+116)^2/29^2 + 190^2/29^2

= (76^2 + 190^2)/29^2

= 41867/29^2 = about 49.793

take the square root

sqr49.793

= about 7.056 = shortest distance from the point to the line

no guarantees these calculations are error free

but that's the basic method

helps to graph the lines and plot the points

and see if the calculated answer looks about right or not

or do a really good graph, draw the line, plot the point

put a compass one end at the point, and sweep out a circle just tangent to the line

that tangent point is the closest point on the line to the given point

from tangent point to the given point is the radius of the circle with center at the given point

use graph paper, if done carefully enough you get a close approximation

or at least a good check on whatever answer you come up with

Or use calculus, optimization method, using derivatives set equal to zero

this problem probably belongs under algebra or geometry more than calculus though

Answer 2

The shortest distance shall be on a line perpendicular to the given line and passing through the given point.

You should draw the line of the given equation and plot the given point to visualize this.

Line perpendicular to given equation:

Slope of given equation is -2 / 5. Slope of perpendicular is 5 / 2

Line must contain (-4, -1)

y = (5 / 2)x + b,

-1 = (5 / 2)(-4) + b

-1 = -10 + b

9 = b

y = (5x / 2) + 9

Determine point of intersection on given equation and equation of perpendicular

(5x / 2) + 9 = (-2x / 5) + 1

The solution shall be a pair of decimal numbers