Answer:
First option.
Third option.
Fifth option.
Step-by-step explanation:
The equation of the line in Slope-Intercept form is:
![y=mx+b](https://tex.z-dn.net/?f=y%3Dmx%2Bb)
Where "m" is the slope and "b" is the y-intercept.
The equation of the line in Point-slope form is:
![y - y_1 = m(x-x_1)](https://tex.z-dn.net/?f=y%20-%20y_1%20%3D%20m%28x-x_1%29)
Where "m" is the slope and
is a point on the line.
By definition, the slopes of perpendicular line are negative reciprocals.
Then, given the line:
![y- 1 = \frac{1}{3}(x+2)](https://tex.z-dn.net/?f=y-%201%20%3D%20%5Cfrac%7B1%7D%7B3%7D%28x%2B2%29)
We know that a line perpendicular to it, must have this slope:
![m=-3](https://tex.z-dn.net/?f=m%3D-3)
Let's check each option:
1) ![y + 2 = -3(x -4)](https://tex.z-dn.net/?f=y%20%2B%202%20%3D%20-3%28x%20-4%29)
Since
, this line is perpendicular to the line ![y- 1 = \frac{1}{3}(x+2)](https://tex.z-dn.net/?f=y-%201%20%3D%20%5Cfrac%7B1%7D%7B3%7D%28x%2B2%29)
2) ![y - 5 = 3(x + 11)](https://tex.z-dn.net/?f=y%20-%205%20%3D%203%28x%20%2B%2011%29)
Since
, this line is not perpendicular to the line ![y- 1 = \frac{1}{3}(x+2)](https://tex.z-dn.net/?f=y-%201%20%3D%20%5Cfrac%7B1%7D%7B3%7D%28x%2B2%29)
3) ![y = -3x-\frac{5}{3}](https://tex.z-dn.net/?f=y%20%3D%20-3x-%5Cfrac%7B5%7D%7B3%7D)
Since
, this line is perpendicular to the line ![y- 1 = \frac{1}{3}(x+2)](https://tex.z-dn.net/?f=y-%201%20%3D%20%5Cfrac%7B1%7D%7B3%7D%28x%2B2%29)
4) ![\frac{1}{3}x - 2](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B3%7Dx%20-%202)
Since
, this line is not perpendicular to the line ![y- 1 = \frac{1}{3}(x+2)](https://tex.z-dn.net/?f=y-%201%20%3D%20%5Cfrac%7B1%7D%7B3%7D%28x%2B2%29)
5) ![3x + y = 7](https://tex.z-dn.net/?f=3x%20%2B%20y%20%3D%207)
Solving for "y":
Since
, this line is perpendicular to the line ![y- 1 = \frac{1}{3}(x+2)](https://tex.z-dn.net/?f=y-%201%20%3D%20%5Cfrac%7B1%7D%7B3%7D%28x%2B2%29)