Greetings and Happy Holidays!
1) Perpendicular to [tex]-x+5y=14[/tex]
In order for lines to be perpendicular, their slopes must be negative reciprocals.
Example of slopes with negative reciprocals: 5 and [tex] \frac{-1}{5} [/tex]
First, rearrange the equation into slope y-intercept form:
[tex]-x+5y=14[/tex]
[tex]5y=x+14[/tex]
[tex] \frac{5y}{5}=\frac{x+14}{5} [/tex]
[tex]y=\frac{1}{5}x+\frac{14}{5}[/tex]
The slope of the equation is: \frac{1}{5}
The negative reciprocal formula: [tex] (m_{1})(m_{2})=-1[/tex]
Solve for the negative reciprocal:
[tex] \frac{1}{5}m_{2}=-1[/tex]
Divide both sides by [tex]\frac{1}{5}[/tex]
[tex]\frac{\frac{1}{5}m_{2} }{ \frac{1}{5}} = \frac{-1}{ \frac{1}{5}}[/tex]
[tex]m_{2}=(-1)(\frac{5}{1})[/tex]
[tex]m_{2}=(\frac{-5}{1})[/tex]
[tex]m_{2}=-5[/tex]
The slope of the new line is: -5
2) Passes through (-5,-2)
Create an equation with the slope discovered in slope y-intercept form.
[tex]y=-5x+b[/tex]
Input the point the line passes through.
[tex](-2)=-5(-5)+b[/tex]
Solve for b (the y-intercept).
[tex]-2=-5(-5)+b[/tex]
Multiply.
[tex]-2=25+b[/tex]
Add -25 to both sides.
[tex](-2)+(-25)=b[/tex]
[tex]-27=b[/tex]
The y-intercept is equal to -27
The Equation of the line is:
[tex]y=-5x-27[/tex]
I hope this helped!
-Benjamin
