Question

A line segment is bisected by line with the equation `6 y - 2 x = 1`. If one end of the line segment is at `(2 ,5 )`, where is the other end?

Answer 1

`(4 1/2, -2 1/2)`

Explanation:

`6y-2x=1`
`6y=2x+1`

`y=1/3x+1/6`
The slope for this equation is `1/3`, therefor the slope for line segment,`m =-3` where `m*m_1=-1`

The equation of line segment is `(y-y_1)=m(x-x_1)` where `x_1=2, y_1=5`

`(y-5)=-3(x-2)`
`y=-3x+6+5=-3x+11` `->a`

so, the intercept between 2 lines is
`1/3x+1/6=-3x+11`

`2/6x+1/6=-3x+11`

`2x+1=6(-3x+11)`
`2x=-18x+66-1`
`20x=65`
`x = 65/20 = 13/4 =3 1/4`

therefore,
`y=-3(13/4)+11`
`y=-39/4+44/4`
`y=5/4=1 1/4`

The line which intercept with both lines is a midpoint of the line segment. Therefore the other end line `(x,y)`

`(x+2)/2=13/4`, `(y+5)/2=5/4`

`x+2=13/2`, `y+5=5/2`

`x=13/2-2`, `y=5/2-5`
`x=9/2=4 1/2`, `y=-5/2=-2 1/2`