Given equation of line are:

y = 3x + 1 ..............1

Slope = 3

    2y = x+3

=> y = x/3 + 3/2 ...........2

Slope = 1/3

y = mx + 4 ........3

Slope = m

Given that equation 1 and 2 are equally inclined with equation 3.

So angle between equation 1 and 3 = angle between equation 2 and 3

=> |(3-m)/(1+3m)| = |(1/2 - m)/(1 + m/2)|

=> |(3-m)/(1+3m)| = |(1 - 2m)/(2 + m)|

=> (3-m)/(1+3m) = (1 - 2m)/(2 + m) and (3-m)/(1+3m) = -(1 - 2m)/(2 + m)

=> (3-m)(2+m) = (1-2m)(1+3m) and (3-m)*(2+m) = -(1+3m)(1-2m)

=> 6+3m-2m-m² = 1+3m-2m-6m² and 6+3m-2m-m² = -(1+3m-2m-6m² )

=> -m² + m + 6 = 1+m-6m² and -m² +m+6 = -1-m+6m²

=> 5m² + 5 = 0 and 7m² -2m-7 = 0 

=> m² + 1= 0

=> m2 = -1 which is not possible.

Again

      7m² -2m-7 = 0

=> m = {2 + √(4 - 4*7*(-7))}/(2*7) and m = {2 - √(4 - 4*7*(-7))}/(2*7)

=> m = {2 + √(4 +4*49)}/14 and m = {2 - √(4+- 4*49)}/14

=> m = {2 + 2√(1 +49)}/14 and m = {2 - 2√(1 + 49)}/14

=> m = {1 + √50}/7 and m = {1 - √50}/7

=> m = {1 + 5√2}/7 and m = {1 - 5√2}/7

So value of m is {1 + 5√2}/7 and m = {1 - 5√2}/7