To begin, we must first solve the for the slope of the original line. Use the formula for slope to do this:

Use the two points we were given:

Reduce the fraction:

We are looking for any answer choice with a slope of . The answer is .

To be totally clear, solve both lines in slope-intercept form:

2x – 4y = 33; –4y = 33 – 2x; y = –33/4 + 0.5x

2x + 4y = 33; 4y = 33 – 2x; y = 33/4 – 0.5x

These lines are definitely not the same. Nor are they parallel—their slopes differ. Likewise, they cannot be perpendicular (which would require not only opposite slope signs but also reciprocal slopes); therefore, they are non-perpendicular intersecting.

There are several ways to solve this problem.  You could solve all of the equations for .  This would give you equations in the form .  All of the lines with the same  value would be parallel.  Otherwise, you could figure out the ratio of  to  when both values are on the same side of the equation.  This would suffice for determining the relationship between the two.  We will take the first path, though, as this is most likely to be familiar to you.

Let's solve each for :

Here, you need to be a bit more manipulative with your equation.  Multiply the numerator and denominator of the  value by :

Therefore, , , and  all have slopes of 

Now, notice that the slope of the line that you have been given is . You know this because slope is merely:

However, for your points, there is no rise at all. You do not even need to compute the value. You know it will be . All lines with slope  are of the form , where  is the value that  has for all  points. Based on our data, this is , for  is always —no matter what is the value for . So, the parallel answer choice is , as both have slopes of .

To begin, solve your equation for . This will put it into slope-intercept form, which will easily make the slope apparent. (Remember, slope-intercept form is , where  is the slope.)

Divide both sides by  and you get:

Therefore, the slope is . Now, you need to test your points to see which set of points has a slope of . Remember, for two points  and , you find the slope by using the equation:

For our question, the pair  and  gives us a slope of :

First write the equation in slope intercept form. Add  to both sides to get . Now divide both sides by  to get . The slope of this line is , so any line that also has a slope of  would be parallel to it. The correct answer is  .

Parallel lines will always have equal slopes. The slope can be found quickly by observing the equation in slope-intercept form and seeing which number falls in the "" spot in the linear equation , 

We are looking for an answer choice in which both equations have the same value. Both lines in the correct answer have a slope of 2, therefore they are parallel.

Lines are parallel when their slopes are the same.

First, we need to place the given equation in the slope-intercept form.

Because the given line has the slope of , the line parallel to it must also have the same slope.