Given the equation of a line, you can find the equation of a line perpendicular to it, knowing the following: The product of the slopes of the lines is -1, it means that:
[tex]m1\cdot m2=-1[/tex]We already know the value of the first slope (which is the slope of the given line). Use this value to find the slope of the perpendicular line:
[tex]\begin{gathered} -\frac{5}{3}\cdot m2=-1 \\ m2=-1\cdot-\frac{3}{5} \\ m2=\frac{3}{5} \end{gathered}[/tex]The slope of the perpendicular line is 3/5. Use this slope and the given point, to find the equation of the perpendicular line using the point slope formula, this way:
[tex]\begin{gathered} y-y1=m(x-x1) \\ y-4=\frac{3}{5}(x-(-5)) \\ y-4=\frac{3}{5}x+3 \\ y=\frac{3}{5}x+7 \end{gathered}[/tex]The equation of the perpendicular line is:
[tex]y=\frac{3}{5}x+7[/tex]To find the equation of the line that is parallel to the given line, use the following information: parallel lines have the same slope, it means:
[tex]m1=m2[/tex]m1 has a value of -5/3, it means the parallel line also has a slope of -5/3.
Use this slope and the given point in the point slope formula, this way:
[tex]\begin{gathered} y-4=-\frac{5}{3}(x-(-5)) \\ y-4=-\frac{5}{3}x-\frac{25}{3} \\ y=-\frac{5}{3}x-\frac{25}{3}+4 \\ y=-\frac{5}{3}x-\frac{13}{3} \end{gathered}[/tex]The equation of the parallel line is:
[tex]y=-\frac{5}{3}x-\frac{13}{3}[/tex]
