The given equation is written in slope-intercept form, and the slope of the line is . The slope of a perpendicular line is the negative reciprocal of the given line. The negative reciprocal here is . Therefore, the correct equation is:

The equation  can be rewritten as follows:

This is the slope-intercept form, and the line has slope . 

The line of the equation  therefore has slope

Since a line perpendicular to this one must have a slope that is the opposite reciprocal of , we are looking for a line with slope .

The slopes of the lines in the four choices are as follows:

; 

; 

: 

:  - this is the correct one.

 can be rewritten as follows:

Any line with equation  is vertical and has undefined slope; a line perpendicular to this is horizontal and has slope 0, and can be written as . The only choice that does not have an  is , which can be rewritten as follows:

This is the correct choice.

The equation  can be rewritten as follows:

This is the slope-intercept form, and the line has slope . 

The line of the equation  has slope

Since a line perpendicular to this one must have a slope that is the opposite reciprocal of  , we are looking for a line that has slope .

The slopes of the lines in the four choices are as follows:

: 

: 

: 

:  - the correct choice.

First, we need to find the slope of the above line. 

The slope of a line. given two points  can be calculated using the slope formula:

Set :

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 3, which would be . Since we want this line to have the same -intercept as the first line, which is the point , we can substitute  and  into the slope-intercept form of the equation:

We calculate the slopes of the lines using the slope formula.

The slope of line  is 

.

The slope of line  is 

.

The lines have the same slope, so either they are distinct parallel lines or one and the same line. One way to check for the latter situation is to find the slope of the line connecting one point on  to one point on  - if the slope is also , the lines coincide. We will use  and :

.

The lines are therefore distinct and parallel.

We calculate the slopes of the lines using the slope formula.

The slope of line  is 

.

The slope of line  is 

.

The lines have the same slope, so either they are distinct, parallel lines or one and the same line. One way to determine which is the case is to find the equations.

Line , the line through  and , has equation

Line , the line through  and , has equation

The lines have the same equation, making them one and the same.

First, we need to find the slope of the above line. 

The slope of a line. given two points  can be calculated using the slope formula:

Set :

A line parallel to this line also has slope . Since it passes through the origin, its -intercept is , and we can substitute  into the slope-intercept form of the equation:

We find the slope of each line by putting each equation in slope-intercept form, , and examining the coefficient of .

 is already in slope-intercept form; its slope is .

To get  in slope-intercept form we solve for :

The slope of this line is .

The slopes are not equal so we can eliminate both "parallel" and "identical" as choices.

Multiply the slopes together:

The product of the slopes of the lines is not , so we can eliminate "perpendicular" as a choice.

The correct response is "neither".

We find the slope of each line by putting each equation in slope-intercept form  and examining the coefficient of .

 is already in slope-intercept form; its slope is .

To get  into slope-intercept form we solve for :

The slope of this line is .

The slopes are not equal so we can eliminate both "parallel" and "one and the same" as choices.

Multiply the two slopes together:

The product of the slopes of the lines is , making the lines perpendicular.