Find the distance between the point (0,0) and the line with the equation -2x-y=-3

(3/5)√5

Step-by-step explanation:

Let's first solve -2x-y=-3 for y.  Add 3 to both sides, obtaining 3 - 2x = y.

Note that if x = 0, y = 3, and so the y-intercept is (0, 3).

But the distance we want here is not that from (0, 0) to (0, 3); rather, it is the distance from (0, 0) measured to the line along a path that is perpendicular to the given line, 3 - 2x = y.  The slope of the perpendicular line is the negative reciprocal of -2, i. e., this slope is 1/2.

Thus, the line along which we measure the distance from (0, 0) is

y = (1/2)x + 0 (since this line passes through the origin).

Now determine the point of intersection of  y = (1/2)x and 3 - 2x = y.  Here we have two equations for y and can equate them to find the value of x:

(1/2)x = 3 - 2x, or, after multiplying all terms by 2, we get:

x = 6 - 4x, or 5x = 6.  Thus, the point of intersection is (6/5, 3 - 2[6/5]), or

(6/5, 3/5).  Find the distance of this point from (0, 0) and we will have answered this question:

The distance from (0, 0) to (6/5, 3/5) is found using the Pythagorean Theorem:

distance = √( [6/5]² + [3/5]² ) = (1/5)√( 45) = (1/5)(3)√5, or (3/5)√5 (answer)

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step-by-step explanation: