Question

Find the point on the line y=x+1 that is closest to the point (4,1).

Answer 1

Do I need to substitute the 4 in place of x and the 1 in place of y? I am not sure how or where to start.

Answer 2

I'll give you a hint. When the point on the line y = x + 1 that is closest to (4,1) is connected to (4,1), the line that is formed is perpendicular to y = x + 1.

Answer 3

so the ending equation would be 1=4+1

Answer 4

or would it be a negative

Answer 5

Uhm, no. First find the line of the perpendicular line that crosses the point (4,1). The slope of a line perpendicular to to another line would be the negative reciprocal of the other line.

The slope of y = x + 1 is 1. The slope of the perpendicular line should be -1. Put together the information you have to get the equation of the perpendicular line.

m = -1, passes (4,1)
y = -1(x - 4) + 1 = 5 - x

Now the perpendicular line intersects the original line, so set the two lines equal.

y = 5 - x
y = x + 1
5 - x = x + 1
x = 2

At x = 2, y = 3. So the closest point is (2,3).