Find the point on the line -3 x + y - 4 =0 which is closest to the point ( -5, 0 ).

find slope of original line

y = 3 x + 4
so slope = m = 3
so slope of line perpendicular to original = m' = -1/m = -1/3
so find line through (-5,0) with slope m' of -1/3
y = -x/3 + b
0 = 5/3 + b
b = -5/3
so y = -x/3 -5/3
3y= -x - 5 or x + 3y + 5 = 0
find intersection of lines
-9x + 3y -12 = 0
+1x + 3y + 5 = 0
------------------
8 x -17 = 0
x = 17/8
find y
then find the difference in distance between the two points
d^2 = change in x^2 + change in y^2

To find the point on the given line that is closest to the point (-5, 0), we can use the concept of distance. The point on the line that is closest to a given point will be the point that has the shortest distance from that given point.

Let's denote the two points on the line and the given point as follows:
Point on the line: (x, y)
Given point: (-5, 0)

The distance between two points can be calculated using the distance formula, which is based on the Pythagorean theorem. The distance formula is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In our case, (x1, y1) = (-5, 0) and (x2, y2) = (x, y). We want to minimize this distance.

Now, we substitute the equation of the line into the distance formula:
d = sqrt((x - (-5))^2 + (y - 0)^2)
= sqrt((x + 5)^2 + y^2)

To find the point on the line that is closest to (-5, 0), we need to minimize the distance formula. Since the distance formula contains a square root, it is easier to minimize the square of the distance.

So we consider:
d^2 = (x + 5)^2 + y^2

Now, we need to express one variable in terms of the other to simplify the expression. From the given line equation, we have:
-3x + y - 4 = 0

Rearranging this equation to express y in terms of x, we get:
y = 3x + 4

Substituting this value of y into the expression for d^2, we have:
d^2 = (x + 5)^2 + (3x + 4)^2

Expanding and simplifying this expression, we get:
d^2 = 10x^2 + 34x + 41

Since we want to minimize d^2, we find the minimum by finding the x-coordinate of the vertex of the quadratic equation 10x^2 + 34x + 41.

The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)

In our case, a = 10, b = 34, and c = 41.

x = -34 / (2 * 10)
x = -34 / 20
x = -1.7

Now that we have the x-coordinate of the point on the line that is closest to (-5, 0), we can substitute this value back into the equation of the line to find the y-coordinate:
y = 3(-1.7) + 4
y = -0.1

Therefore, the point on the line -3x + y - 4 = 0 that is closest to (-5, 0) is (-1.7, -0.1).