Given the equation of the line:
[tex]y=-5x-\frac{1}{2}[/tex]• You can identify that it is written in Slope-Intercept Form:
[tex]y=mx+b[/tex]Where "m" is the slope of the line, and "b" is the y-intercept.
Notice that:
[tex]\begin{gathered} m_1=-5 \\ b_1=-\frac{1}{2} \end{gathered}[/tex]• By definition, parallel lines have the same slope, but their y-intercepts are different.
Therefore, you can determine that the slope of the line parallel to the first line is:
[tex]m_2=-5[/tex]You know that this line passes through this point:
[tex](-4,2)[/tex]Therefore, substituting the slope and the coordinates of that point into this equation:
[tex]y=m_2x+b_2[/tex]And solving for the y-intercept, you get:
[tex]\begin{gathered} 2=(-5)(-4)+b_2 \\ \\ 2-20=b_2 \\ \\ b_2=-18\frac{}{} \end{gathered}[/tex]Then, the equation of the line parallel to the first line is:
[tex]y=-5x-18[/tex]• By definition, the slopes of perpendicular lines are opposite reciprocal, therefore, the slope of this line is:
[tex]m_3=\frac{1}{5}[/tex]Using the same procedure used before to find the y-intercept, you get:
[tex]\begin{gathered} 2=(\frac{1}{5})(-4)+b_3 \\ \\ 2+\frac{2}{5}=b_3 \\ \\ b_3=\frac{14}{5} \end{gathered}[/tex]Therefore, its equation is:
[tex]y=\frac{1}{5}x+\frac{14}{5}[/tex]Hence, the answer is:
- Equation for the parallel line:
[tex]y=-5x-18[/tex]- Equation for the perpendicular line:
[tex]y=\frac{1}{5}x+\frac{14}{5}[/tex]