### All GRE Math Resources

## Example Questions

### Example Question #1 : How To Find Out If Lines Are Parallel

There are two lines:

2x – 4y = 33

2x + 4y = 33

Are these lines perpendicular, parallel, non-perpendicular intersecting, or the same lines?

**Possible Answers:**

The same

Non-perpendicular intersecting

Perpendicular

None of the other answers

Parallel

**Correct answer:**

Non-perpendicular intersecting

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.

### Example Question #1 : How To Find Out If Lines Are Parallel

Which of the above-listed lines are parallel?

**Possible Answers:**

All four lines

and

, , and

and

None of them

**Correct answer:**

, , and

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

### Example Question #2 : How To Find Out If Lines Are Parallel

Which of the following is parallel to the line passing through and ?

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #3 : How To Find Out If Lines Are Parallel

Which of the following is parallel to ?

**Possible Answers:**

The line between the points and

The line between the points and

The line between the points and

The line between the points and

The line between the points and

**Correct answer:**

The line between the points and

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 :

### Example Question #21 : Coordinate Geometry

Which of the following lines is parallel to:

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #1 : How To Find Out If Lines Are Parallel

Which pair of linear equations represent parallel lines?

**Possible Answers:**

**Correct answer:**

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.

### Example Question #3 : How To Find Out If Lines Are Parallel

Which of the following equations represents a line that is parallel to the line represented by the equation ?

**Possible Answers:**

**Correct answer:**

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.

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