Please help me solve this.

Question

Find the point on the line 1x + 1y - 5 = 0 which is closest to the point (-1,−1).

Answer 1

Recall that the shortest distance between a line and a point is its perpendicular distance. So, the line perpendicular to x+y-5=0 at (1,1) is y=x, so we can just find x and y with simple substitution:

x+y-5=0

x+x-5=0

2x-5=0

2x=5

x=2.5

Since y=x, then y=2.5 as well, which means the point that is closest to (-1,-1) on the line x+y-5=0 is (2.5,2.5)

Hope this helped!

Answer 2

If we rewrite the equation, we will get: y = 5 - x

What is the distance between any point on that line and (-1,-1)? We can use the distance formula.

d = √[(x₂ - x₁)² - (y₂ - y₁)²]

x₁ = -1

x₂ = -1

y₁ = x

y₂ = 5 - x

Now you have a square root expression in terms of "x" only. Take the derivative, set it equal to 0 to find your critical points, and use the first derivative test. Once you find the "x" that minimizes "d" (where the sign goes from negative to positive), plug that into (5 - x) to find the exact point on y = 5 - x that is closest to (-1,-1).