Answer:
The equation of a line perpendicular to the line that passes through the given point as;
[tex]y=\frac{x}{5}-\frac{26}{5}[/tex]
Explanation:
Given that we want to find the eqution of the line perpendicular to the line equation below;
[tex]y=-5x-\frac{4}{3}[/tex]
Whose slope is;
[tex]m_1=-5[/tex]
For two lines to be perpendicular, the slope must follow the rule below;
[tex]\begin{gathered} m_1.m_2=-1 \\ m_2=\frac{-1}{m_1} \end{gathered}[/tex]
Substituting the value of the given slope;
[tex]\begin{gathered} m_2=\frac{-1}{m_1}=\frac{-1}{-5} \\ m_2=\frac{1}{5} \end{gathered}[/tex]
Now we have the slope of our line.
We can now derive the equation using the point-slope equation of line;
[tex]y-y_1=m(x-x_1)[/tex]
With the given point;
[tex](x_1,y_1)=(6,-4)[/tex]
Substituting the slope and the given point, we have;
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y-(-4)=\frac{1}{5}(x-6) \\ y+4=\frac{x}{5}-\frac{6}{5} \\ y=\frac{x}{5}-\frac{6}{5}-4 \\ y=\frac{x}{5}-\frac{26}{5} \end{gathered}[/tex]
Therefore, we have the equation of a line perpendicular to the line that passes through the given point as;
[tex]y=\frac{x}{5}-\frac{26}{5}[/tex]
