Question

A line segment is bisected by a line with the equation `4 y - 3 x = 2`. If one end of the line segment is at `( 2 , 1 )`, where is the other end?

Answer 1

`(1 1/25,2 7/25)`

Explanation:

`4y-3x=2`, `4y=3x+2`
`y=3/4x+1/2` `->a`

gradient of the other line, `m = -1/(3/4)=-4/3`

The equation of the line thru `(2,1)`,

`y-y_1=m(x-x_1)`
`y-1=-4/3(x-2)`
`y=-4/3x+8/3+1`
`y=-4/3x+11/3`

replacing y from `a`

`3/4x+1/2=-4/3x+11/3`

`3/4x+4/3x=11/3-1/2`

`9/12x+16/12x=22/6-3/6`

`25/12x=19/6`

`x=19/6*12/25=38/25`

`y=3/4(38/25)+1/2`

`y=57/50+25/50=82/50=41/25`

`(38/25,41/25)` is a midpoint of the line.

The other end of the line, `(x,y)`.

`(x+2)/2=38/25`
`x=38/25*2-2`
`x=76/25-50/25=26/25=1 1/25`

`(y+1)/2=41/25`
`y=41/25*2-1`
`y=82/25-25/25=57/25=2 7/25`

The other end `(1 1/25,2 7/25)`