The slope intercept form of a line can be expressed as,
[tex]y=mx+c[/tex]Here, m is the slope of the line and c is the y intercept.
Comparing the above equation with the given equation of a line y=-5x+2, we get
m=-5.
The slope of a line perpendicular to line with slope m is -1/m.
Hence, the slope of line perpendicular to y=-5x+2 is,
[tex]m_1=\frac{-1}{m}=\frac{-1}{-5}=\frac{1}{5}[/tex]The new line is given to be passing through point with coordinates (x1, y1)=(3, -1).
The point slope form of a line passing through point with coordinates (x1, y1)=(3, -1) and having slope m1 is,
[tex]\begin{gathered} y-y_1=m_1(x-x_1) \\ y-(-1)=\frac{1}{5}(x-3) \\ y+1=\frac{1}{5}x-\frac{3}{5} \\ y=\frac{1}{5}x-\frac{3}{5}-1 \\ y=\frac{1}{5}x-\frac{3-5}{5} \\ y=\frac{1}{5}x-\frac{8}{5} \end{gathered}[/tex]Therefore, the slope-intercept form of the equation of the line perpendicular to y = -5x + 2 and passing through (3,-1) is,
[tex]y=\frac{1}{5}x-\frac{8}{5}[/tex]