A line segment is bisected by a line with the equation - 4 y + x = 1 . If one end of the line segment is at ( 7 , 2 ), where is the other end?

1 Answer
Douglas K.
Nov 30, 2016

The other end is at (123/17, 18/17)

Explanation:

Let's write the given line in the form ax + by = c

x - 4y = 1" [1]"

The general equation of lines perpendicular to this line is:

4x + y = c

To find the value of c, substitute 7 for x and 2 for y:

4(7) + 2 = c

c = 30

The equation of the bisected line segment is:

4x + y = 30" [2]"

The midpoint is at the intersection of of these two lines:

x - 4y = 1" [1]"
4x + y = 30" [2]"

Multiply equation [1] by 4 and subtract from equation [2]

17y = 26

y = 26/17

Let Deltay = the change in the y coordinate = 26/17 - 2

The y coordinate of the other end of the line, y_1 will have 2 twice the change from the starting y coordinate:

y_1 = 2Deltay + y_0

y_1 = 2(26/17 - 2) + 2

y_1 = 52/17 - 2

y_1 = 18/17

To find the corresponding x coordinate, substitute 18/17 for y into equation [2]:

4x + 18/17 = 30

4x = 30 - 18/17

x_1 = 123/17

The other end is at (123/17, 18/17)

Here is a graph of two lines and two points:

![Desmos.com](useruploads.socratic.org)

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