Linear Algebra question

Question

Find the point on the line y+2x+1 that is closest to the point (3,5)

Answer 1

If your question is supposed to say "Find the point on the line y = 2x + 1 that is closest to the point (3,5)", then:

To find the point, write an equation for the line perpendicular to y = 2x + 1 that goes through the point (3, 5). Then, find where that line intersects y = 2x + 1 and you have the point.

The line perpendicular to y = 2x + 1 would have a slope of -1/2. You can then use the point-slope form to write the equation of the line with slope -1/2 that passes through (3, 5):

y - 5 = -1/2(x - 3)

y - 5 = -1/2x + 3/2

y = -1/2x + 3/2 + 5

y = -1/2x + 3/2 + 10/2

y = -1/2x + 13/2

Now, to find the intersection point of:

y = 2x + 1 and

y = -1/2x + 13/2

set "2x + 1" equal to "-1/2x + 13/2"

2x + 1 = -1/2x + 13/2

2.5x = 13/2 - 1

2.5x = 13/2 - 2/2

2.5x = 11/2

2.5x = 5.5

x = 5.5/2,5

x = 2.2

Then solve for y:

y = 2x + 1

y = 2(2.2) + 1

y = 5.4

So the point on the line y = 2x + 1 that is closest to (3.5) is (2.2, 5.4)