Question

Explain in words how to write an equation that is part one:parallel and then also part two: perpendicular to the equation y= 2/3x-4 passing through the point (-2,-5) write your answer in standard form

Answer 1

You are given the line l with equation
`y=(2)/(3)x-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
`(2)/(3).`

Then the equation is

`y=(2)/(3)x+a.`

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=(2)/(3)\cdot (-2)+a,\\ \\a=-5+(4)/(3)=(-15+4)/(3)=-(11)/(3).`

Thus, the equation of parallel line is

`y=(2)/(3)x-(11)/(3).`

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.`

Therefore, the slope of perpendicular line is

`(2)/(3)\cdot m_2=-1,\\ \\m_2=-(3)/(2).`

Then the equation is

`y=-(3)/(2)x+b.`

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=-(3)/(2)\cdot (-2)+b,\\ \\b=-5-3=-8.`

Thus, the equation of perpendicular line is

`y=-(3)/(2)x-8.`