find the point on the line y=2x-3 that is nearest to the origin?

Answer 1

(0,-3)
Make a chart of x coordinates. I used -2,-1, 0, 1, 2

Then substitue each x in the problem to get y
ex. y=2(-2)-3
y=-7 and x=-2
your cordinate is (-2,-7)
Continue until u find the point closet to 1
y=2(1)-3
y=-1
your coordinate and answer is (1,-1)

Answer 2

Okay, this may be a bit long winded, but here goes.

The problem is harder than it seems at first blush.

The equation is a simple linear expression, in which the y-intercept lies at -3, and the x-interpect is at 1.5 (and the slope is +2).

Okay, so that said we need to find the point closest to the origin. To do that, it is useful to note that we are basically going to find the point at which the radius from the origin is at it's minimum.

The equation relating the radius to x and y is:

r = sqrt(x^2 + y^2)

Substitution yields:
r = sqrt (x^2 + (2x - 3)^2)

Reducing yields:
r = sqrt(5x^2-12x+9)

Notice, the when the argument inside the square-root is smallest, we have found the minimum radius from the origin.

To find the minimum we take the derivative of (5x^2-12x+9) with respect to x and set it to zero.

d/dx (5x^2-12x+9) = 10x - 12
10x - 12 = 0
Thus, x=12/10 = 1.2

y = 2x-3 = 2(1.2)-3 = -0.6

So, the point at which the line is closest to the origin is (1.2,-0.6)

Good Luck to you!

Answer 3

The point in question will be the intersection of the given line and a line perpendicular to it that passes thru the origin.

The equation of the given line is:

y = 2x - 3

The slope m of the perpendicular line is the negative reciprocal of the given line. The slope of the given line is 2. The slope of the perpendicular line is:

m = -1/2

The equation of the perpendicular line is:

y - 0 = (-1/2)(x - 0)
y = -x/2

Set the two equations equal.

2x - 3 = -x/2
5x/2 = 3
x = 3(2/5) = 1.2

y = -x/2 = -1.2/2 = -0.6

The closest point on the line to the origin is (1.2, -0.6).

Answer 4

I reckon its (1.5,0). The line intercepts x axis at (1.5,0) and y-axis (0,-3) . so I reckon nearest point to the origin will be (1.5,0).

Answer 5

(1,-1)