If the lines `y = 3x + 1` and `2y = x + 3` are equally inclined to the line `y = m x + 4` , find the value of m.

Explanation

`y=3x+1` and `2y=x+3`
Slope `m_(1)=3` and `m_(2)=(1)/(2)`
Now, the slope of line `y=mx+4` is `m`.
`:'` This straight line makes equal angles with the given two lines,
`(m_(1)-m)/(1+m_(1)m)=(m-m_(2))/(1+mm_(2))`
`implies (3-m)/(1+3m)=(m-(1)/(2))/(1+(1)/(2)m)`
`=(2m-1)/(2+m)`
`implies(1+3m)(2m-1)=(3-m)(2+m)`
`implies 6m^(2)-m-1=6+m-m^(2)`
`implies 7m^(2)-2m-7=0`
`implies m=(2+-sqrt(4+196))/(14)`
`=(2+-10sqrt(2))/(14)`
`=(1+-5sqrt(2))/(7)`