You are given the line l with equation ![y=\dfrac{2}{3}x-4.](https://tex.z-dn.net/?f=y%3D%5Cdfrac%7B2%7D%7B3%7Dx-4.)
1. The equation of line that passes through the point (-2,-5) and is parallel to the line l.
Parallel lines have the same slope. So the slope of unknown line is ![\dfrac{2}{3}.](https://tex.z-dn.net/?f=%5Cdfrac%7B2%7D%7B3%7D.)
Then the equation is
![y=\dfrac{2}{3}x+a.](https://tex.z-dn.net/?f=y%3D%5Cdfrac%7B2%7D%7B3%7Dx%2Ba.)
This line passes through point (-2,-5), this means that coordinates of this point satisfy the equation, substitute x=-2 and y=-5 into equation:
![-5=\dfrac{2}{3}\cdot (-2)+a,\\ \\a=-5+\dfrac{4}{3}=\dfrac{-15+4}{3}=-\dfrac{11}{3}.](https://tex.z-dn.net/?f=-5%3D%5Cdfrac%7B2%7D%7B3%7D%5Ccdot%20%28-2%29%2Ba%2C%5C%5C%20%5C%5Ca%3D-5%2B%5Cdfrac%7B4%7D%7B3%7D%3D%5Cdfrac%7B-15%2B4%7D%7B3%7D%3D-%5Cdfrac%7B11%7D%7B3%7D.)
Thus, the equation of parallel line is
![y=\dfrac{2}{3}x-\dfrac{11}{3}.](https://tex.z-dn.net/?f=y%3D%5Cdfrac%7B2%7D%7B3%7Dx-%5Cdfrac%7B11%7D%7B3%7D.)
2. The equation of line that passes through the point (-2,-5) and is perpendicular to the line l.
Perpendicular lines have slopes that satisfy the condition
![m_1\cdot m_2=-1.](https://tex.z-dn.net/?f=m_1%5Ccdot%20m_2%3D-1.)
Therefore, the slope of perpendicular line is
![\dfrac{2}{3}\cdot m_2=-1,\\ \\m_2=-\dfrac{3}{2}.](https://tex.z-dn.net/?f=%5Cdfrac%7B2%7D%7B3%7D%5Ccdot%20m_2%3D-1%2C%5C%5C%20%5C%5Cm_2%3D-%5Cdfrac%7B3%7D%7B2%7D.)
Then the equation is
![y=-\dfrac{3}{2}x+b.](https://tex.z-dn.net/?f=y%3D-%5Cdfrac%7B3%7D%7B2%7Dx%2Bb.)
This line passes through point (-2,-5), this means that coordinates of this point satisfy the equation, substitute x=-2 and y=-5 into equation:
![-5=-\dfrac{3}{2}\cdot (-2)+b,\\ \\b=-5-3=-8.](https://tex.z-dn.net/?f=-5%3D-%5Cdfrac%7B3%7D%7B2%7D%5Ccdot%20%28-2%29%2Bb%2C%5C%5C%20%5C%5Cb%3D-5-3%3D-8.)
Thus, the equation of perpendicular line is
![y=-\dfrac{3}{2}x-8.](https://tex.z-dn.net/?f=y%3D-%5Cdfrac%7B3%7D%7B2%7Dx-8.)