In order to determine whether the equations are parallel, perpendicular, or neither, let's simply each equation into a slope-intercept form or basically, solve for y.
Let's start with the first equation.
[tex]\frac{6x-5y}{2}=x+1[/tex]Cross multiply both sides of the equation.
[tex]6x-5y=2(x+1)[/tex][tex]6x-5y=2x+2[/tex]Subtract 6x on both sides of the equation.
[tex]6x-5y-6x=2x+2-6x[/tex][tex]-5y=-4x+2[/tex]Divide both sides of the equation by -5.
[tex]-\frac{5y}{-5}=\frac{-4x}{-5}+\frac{2}{-5}[/tex][tex]y=\frac{4}{5}x-\frac{2}{5}[/tex]Therefore, the slope of the first equation is 4/5.
Let's now simplify the second equation.
[tex]-4y-x=4x+5[/tex]Add x on both sides of the equation.
[tex]-4y-x+x=4x+5+x[/tex][tex]-4y=5x+5[/tex]Divide both sides of the equation by -4.
[tex]\frac{-4y}{-4}=\frac{5x}{-4}+\frac{5}{-4}[/tex][tex]y=-\frac{5}{4}x-\frac{5}{4}[/tex]Therefore, the slope of the second equation is -5/4.
Since the slope of each equation is the negative reciprocal of each other, then the graph of the two equations is perpendicular to each other.