Find the shortest distance between the point and the line (pre-calculus)

find the shortest distance between the point and the line

Y= (2/7x)-(1/4) Point (5,-6)

pls and thank you

Equation of the line:
y = (2/7)x - (1/4)
28y = 8x - 7
8x - 28y - 7 = 0
Shortest distance of point (5,-6) to the line:
D = | 8*(5) - 28*(-6) - 7| / √(8² + 28²)
. .= | 40 + 168 - 7 | / √(100)
. .= 201 / √(212)
. .= 201 / 4√(53)

Technically the equation you wrote is not a line. I suspect you meant Y = (2/7)X - (1/4). That nitpick aside, what you are minimizing is the squared-distance between (X, (2/7)X - (1/4) and (5, -6). You can write this out as D^2(X) = (X - 5)^2 + ((2/7)X - (1/4) + 6)^2.

Multiply this all out and get a quadratic polynomial in X, which you can minimize by finding the derivative and setting it equal to zero.

--OR--

You can find the line with slope -7/2 which goes thru (5,-6). This line is guaranteed to be perpendicular to the given line, and from geometry you know that the perpendicular line is the shortest distance. Solve the pair of linear equations to find the intersection point Z, and the distance from Z to (5,-6) is the answer.

First we'll find the equation of the line perpendicular to y = (2/7)x - (1/4) through the point (5, -6)

The slope of the perpendicular is - 7/2, so y = (-7/2)x + b; when x = 5, y = -6.

-6 = (-7/2)(5) + b......................b = 23/2, so the perpendicular is y = (-7/2)x + 23/2

Now we have to find the point where the two lines intersect.

(2/7)x - 1/4 = - 7/2 x + 23/2.......53/14 x = 47/4........x = 658/212 = 329/106.

Now you need to find the corresponding y value. Then use the distance formula to find the distance between (5, -6) and (329/106, y).