We're starting with the line
[tex]5x + 7y = 8[/tex]
All the lines parallel to it will be of the form
[tex]5x + 7y = \textrm{constant}[/tex]
and when our intersection point is (5,-3) the constant must be
[tex]5x + 7y = 5(5)+7(-3) = 25 - 21 = 4[/tex]
That's the answer but we have to turn this into slope intercept form so we solve for y:
[tex]7y = -5x + 4[/tex]
[tex]y = - \frac 5 7 x + \frac 4 7[/tex]
Answer: y = (-5/7)x + 4/7
For perpendicular lines, we swap the coefficients on x and y, negating one:
[tex]7x - 5y = \textrm{constant}[/tex]
Again the point gives the constant:
[tex]7x - 5y = 7(5) - 5(-3) = 35 + 15 = 50[/tex]
Solve for y for slope/intercept:
[tex]5y = 7x - 50[/tex]
[tex]y = \frac 7 5 x - 10
Answer: y = (7/5) x - 10
Let's check with a nice plot:
