Question

If given an equation of a line such as y = (-1/2)x + 6, demonstrate how to create an equation of a line parallel and perpendicular to this line that goes through another point such as (4, 10).

Answer 1

If line $y_{1}=m_{1}x+b_{1}$ is parallel to line $y_{2}=m_{2}x+b_{2}$ i.e $y_{1} \parallel y_{2}$ then, by definition, $m_{2}=m_{1}$. Taking this along with the given point $(h,y)$ a line can be constructed in point slope form that satisfies the requirements as $y_{2}-k=m_{2}(x-h)$.

Similarly, If line $y_{1}=m_{1}x+b_{1}$ is perpendicular to line $y_{2}=m_{2}x+b_{2}$ i.e $y_{1} \perp y_{2}$ then, by definition, $m_{2}=-\frac{1}{m_{1}$. Taking this along with the given point $(h,y)$ a line can be constructed in point slope form that satisfies the requirements as $y_{2}-k=m_{2}(x-h)$. .