Question

How do you: Find the shortest distance, to two decimal places from the origin to the line x - 2y = - 8 without using the formula.

Finding the shortest distance, to two decimal places, from the point P ( 7,5) to the line 2x+3y = 18?

Thanks

The square of the distance from a point on that line, at location x, is
D^2 = (7-x)^2 + (5-y)^2
= (7-x)^2 + [-1 + (2x/3)]^2
Set the derivative dD^2/dx = 0
Solve for x and then use the equation 2x+3y = 18 to get y. Then compute D

Another way to do this is to find the equation for the line through the point P that is perpendicular to the line. Since the slope of 2x+3y = 18 is -2/3, the slope of the perpendicular is 3/2.

another way would be to use vectors.

Have you studied vectors?
There is no point to show you the method unless you know the topic.