Answer:
[tex]\displaystyle y - 8 = -\frac{2}{5}\, (x + 7)[/tex] is perpendicular to [tex]\displaystyle y - 4 = \frac{5}{2} \, (x + 3)[/tex] and goes through the point [tex](-7,\, 8)[/tex].
Step-by-step explanation:
Consider a line that has a slope of [tex]m[/tex] and goes through the point [tex](x_{0},\, y_{0})[/tex]. The point-slope equation of this line would be:
[tex]y - y_{0} = m\, (x - x_{0})[/tex].
The equation of "the given line" in this question is in the point-slope form. Compare the equation of this given line to [tex]y - y_{0} = m\, (x - x_{0})[/tex]. [tex]m = (5/2)[/tex].
The coefficient of [tex]x[/tex] would be [tex]m = (5 / 2)[/tex]. In other words, the slope of this given line would be [tex](5/2)[/tex].
If two lines are perpendicular to one another, the product of their slopes would be [tex](-1)[/tex].
Since the slope of the given line is [tex](5 / 2)[/tex], the slope of a line perpendicular to this line would be:
[tex]\displaystyle \frac{-1}{5 / 2} = -\frac{2}{5}[/tex].
The question requested that this line should go through the point [tex](-7,\, 8)[/tex]. Since the slope of that line is found to be [tex](-2/5)[/tex], the point-slope equation of that line would be:
[tex]\displaystyle y - 8 = -\frac{2}{5}\, (x - (-7))[/tex].
Simplify this equation to get:
[tex]\displaystyle y - 8 = -\frac{2}{5}\, (x + 7)[/tex].
