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.

In order to move from the lower left plotted point to the upper right plotted point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. The slope is the rise-to-run ratio, so the slope of the line is . Any line parallel to this line will have the same slope, so the correct response is .

In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. The slope of a line is the ratio of rise to run, so the slope of the line shown is .

A line perpendicular to this will have a slope equal to the opposite of the reciprocal of . This is .

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 2, which would be .

Since we want the line to have the same -intercept as the above line, which is the point , we can use the slope-intercept form to help us.  We set 

, and solve for :

Substitute for  and  in the slope-intercept form, and the equation is

.

The equation  can be rewritten as follows:

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

The line  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 that has slope .

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

: 

: 

: 

:   This is the correct choice.

Subtract  from both sides.

The slope of this line is negative three.

The slope of the perpendicular line is the negative reciprocal of this slope.

The answer is:  

Two lines are parallel if they have the same slope. Now, we know the slope-intercept form is written as follows:

where m is the slope and b is the y-intercept. Now, given the equation

we can see the slope is -3. So, to find a line parallel to this line, we will have to find an equation that also have a slope of -3. 

If we look at the equation

we can see it has a slope of -3. Therefore, this equation is parallel to the original equation.