perpendicularWe were given the following information:
The equation of a line is given by: y = - 1/4 x + 9
We want to obtain the equation for a line perpendicular to this line & that passes through the point (4, -8). This is shown below:
The general equation of a straight line is given by:
[tex]\begin{gathered} y=mx+b \\ where\colon \\ m=slope \\ b=y-intercept \end{gathered}[/tex]The equation of the line given us is:
[tex]\begin{gathered} y=-\frac{1}{4}x+9 \\ \text{Comparing this with the general equation, we will deduce that:} \\ mx=-\frac{1}{4}x \\ m=-\frac{1}{4} \\ \text{Thus, the slope of this line is: }-\frac{1}{4} \end{gathered}[/tex]The relationship between the slope of a line and the slope of a line perpendicular to it is given by the statement "the product of the slopes of two lines perpendicular to one another is negative one"
This is expressed below:
[tex]\begin{gathered} m\times m_{perpendicular}=-1 \\ m_{perpendicular}=-\frac{1}{m} \\ m=-\frac{1}{4} \\ m_{perpendicular}=\frac{-1}{-(\frac{1}{4})} \\ m_{perpendicular}=4 \\ \\ \therefore m_{perpendicular}=4 \end{gathered}[/tex]Therefore, the slope of the perpendicular line is: 4
We were told that the perpendicular line passes through the point (4, -8). We will obtain the equation of the perpendicular line using the Point-Slope equation. This is shown below:
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ (x_1,y_1)=(4,-8) \\ m\Rightarrow m_{perpendicular}=4_{} \\ \text{Substitute the values of the variables into the initial equation above, we have:} \\ y-\mleft(-8\mright)=4(x-4) \\ y+8=4(x-4) \\ y+8=4x-16 \\ \text{Subtract ''8'' from both sides, we have:} \\ y=4x-16-8 \\ y=4x-24 \\ \\ \therefore y=4x-24 \end{gathered}[/tex]The graphical representation of this is given below:
