<u>Given</u>:
Given that the graph of the equation of the line.
The line that is perpendicular to the given line and passes through the point (2,-1)
We need to determine the equation of the line perpendicular to the given line.
<u>Slope of the given line:</u>
The slope of the given line can be determined by substituting any two coordinates from the line in the slope formula,
![m=\frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
Substituting the coordinates (-1,3) and (2,2), we get;
![m_1=\frac{2-3}{2+1}](https://tex.z-dn.net/?f=m_1%3D%5Cfrac%7B2-3%7D%7B2%2B1%7D)
![m_1=-\frac{1}{3}](https://tex.z-dn.net/?f=m_1%3D-%5Cfrac%7B1%7D%7B3%7D)
Thus, the slope of the given line is ![m_1=-\frac{1}{3}](https://tex.z-dn.net/?f=m_1%3D-%5Cfrac%7B1%7D%7B3%7D)
<u>Slope of the perpendicular line:</u>
The slope of the perpendicular line can be determined by
![m_2=-\frac{1}{m_1}](https://tex.z-dn.net/?f=m_2%3D-%5Cfrac%7B1%7D%7Bm_1%7D)
Substituting
, we get;
![m_2=-\frac{1}{-\frac{1}{3}}](https://tex.z-dn.net/?f=m_2%3D-%5Cfrac%7B1%7D%7B-%5Cfrac%7B1%7D%7B3%7D%7D)
simplifying, we get;
![m_2=3](https://tex.z-dn.net/?f=m_2%3D3)
Thus, the slope of the perpendicular line is 3.
<u>Equation of the perpendicular line:</u>
The equation of the perpendicular line can be determined using the formula,
![y-y_1=m(x-x_1)](https://tex.z-dn.net/?f=y-y_1%3Dm%28x-x_1%29)
Substituting
and the point (2,-1) in the above formula, we have;
![y+1=3(x-2)](https://tex.z-dn.net/?f=y%2B1%3D3%28x-2%29)
![y+1=3x-6](https://tex.z-dn.net/?f=y%2B1%3D3x-6)
![y=3x-7](https://tex.z-dn.net/?f=y%3D3x-7)
Thus, the equation of the perpendicular line is ![y=3x-7](https://tex.z-dn.net/?f=y%3D3x-7)
Hence, Option d is the correct answer.