[tex]\begin{gathered} \text{Parallel line} \\ P(4,-2) \\ 4x+7y=9 \\ 7y=9-4x \\ y=\frac{-4x}{7}+\frac{9}{7} \\ Two\text{ lines are parallel when they have the same slope} \\ y=\frac{-4x}{7}+b\text{ } \\ \text{with the point given we can calculate b} \\ -2=\frac{-4\cdot(4)}{7}+b\text{ } \\ -2=\frac{-16}{7}+b \\ b=-2+\frac{16}{7} \\ b=\frac{2}{7} \\ y=\frac{-4}{7}x+\frac{2}{7} \\ 7y=-4x+2 \\ 7y+4x=2 \\ the\text{ equation for the parallel line is }4x+7y=2 \\ \text{perpendicular line} \\ P(4,-2) \\ 4x+7y=9 \\ 7y=9-4x \\ y=\frac{-4x}{7}+\frac{9}{7},\text{ slope 1 =}\frac{-4}{7} \\ Two\text{ lines are perpendicular when you multiply both slopes} \\ and\text{ you get -1} \\ -1=slope\text{ 1 }\cdot\text{ slope 2} \\ slope\text{ 2 =}\frac{-1}{\text{slope1}} \\ \\ slope\text{ 2 =}\frac{-1}{(\frac{-4}{7})} \\ \\ slope\text{ 2 =}\frac{7}{4} \\ y=\frac{7}{4}x+b \\ P(4,-2) \\ -2=\frac{7}{4}\cdot(4)+b \\ -2=7+b \\ b=-2-7 \\ b=-9 \\ y=\frac{7}{4}x-9 \\ -\frac{7}{4}x+y=-9 \\ \\ the\text{ equation for the perpendicular line is }-\frac{7}{4}x+y=-9 \end{gathered}[/tex]
