A line segment is bisected by a line with the equation `4 y - 2 x = 3`. If one end of the line segment is at `( 5 , 6 )`, where is the other end?

First of all, the bisector of a line segment is perpendicular to the segment.

Perpendicular slope is the negative reciprocal of the original slope, so first step is to find the slope of the given line.

Step 1: Finding the equation of the line segment

Convert to slope-intercept form `y=mx+b`

`4y-2x=3`
`4y = 2x+3`
`y=2/4x+3/4`
`color(red)(m=2/4)`

Taking the negative reciprocal we have the slope of our line segment `color(red)(m=-4/2)`

NOTE: Because the bisector intersects the line segment at the midpoint, we can use the given point to find our `b` value.

`6=-4/2(5)+b`
`6=-20/2+b`
`6+10=b`
`color(red)(b=16)`

This gives us the equation of the line segment on infinite domain `y=-4/2x+16`

Step 2: Equate the two lines to find the intersection point at x

`2/4x+3/4=-4/2x+16`

Group x values together and factor

`x(2/4+4/2)=-3/4+16`
`x(2/4+8/4)=-3/4+64/4`

`(10x)/4=61/4`

`10x=61`

`color(red)(x=6.1)`

Sub in 6.1 to either equation to find y.

`y=-4/2(6.1)+16`
`y=-24.4/2+16`
`y=-12.2+16`
`y=3.8`

So our intersection point is at `(6.1,3.8)`

Step 3: Find the endpoint using the midpoint formula

Because the intersection point `(6.1,3.8)` is also a midpoint, we can find the final point using the midpoint formula.

`((x_1+x_2)/2, (y_1+y_2)/2)`

Substitute the original points at `x_1,y_1`

`(5+x_2)/2=6.1`
`x_2=6.1(2)-5`
`x_2=7.2`

`(6+y_2)/2=3.8`
`y_2=3.8(2)-6`
`y_2=1.6`

Answer

`(x,y)=(7.2,1.6)`