### All GRE Math Resources

## Example Questions

### Example Question #1 : Parallel Lines

What is one possible equation for a line parallel to the one passing through the points (4,2) and (15,-4)?

**Possible Answers:**

y = 6/11x - 33

y = -6/11x + 57.4

y = 15x + 12

y = 11/6x + 88

y = -11/6x + 8.32

**Correct answer:**

y = -6/11x + 57.4

(4,2) and (15,-4)

All that we really need to ascertain is the slope of our line. So long as a given answer has this slope, it will not matter what its y-intercept is (given the openness of our question). To find the slope, use the formula: m = rise / run = (y1 - y2) / (x1 - x2):

(2 - (-4)) / (4 - 15) = (2 + 4) / -11 = -6/11

Given this slope, our answer is: y = -6/11x + 57.4

### Example Question #2 : Parallel Lines

Lines m and n are parallel

What is the value of angle ?

**Possible Answers:**

145

125

115

130

180

**Correct answer:**

145

By using the complementary and supplementary rules of geometry (due to lines m and n being parallel), as well as the fact that the sum of all angles within a triangle is 180, we can carry through the operations through stepwise subtraction of 180.

x = 125 → angle directly below also = 125. Since a line is 180 degrees, 180 – 125 = 55. Since right triangle, 90 + 55 = 145 → rightmost angle of triangle 180 – 145 = 35 which is equal to the reflected angle. Use supplementary rule again for 180 – 35 = **145 = y**.

Once can also recognize that both a straight line and triangle must sum up to 180 degrees to skip the last step.

### Example Question #3 : Parallel Lines

What is the equation for the line running through and parallel to ?

**Possible Answers:**

**Correct answer:**

To begin, solve the given equation for . This will give you the slope-intercept form of the line.

Divide everything by :

Therefore, the slope of the line is .

Now, for a point , the point-slope form of a line is:

, where is the slope

For our point, this is:

This is the same as:

Distribute and solve for :

### Example Question #1 : How To Find The Equation Of Parallel Lines

What is the equation for the line running through and parallel to ?

**Possible Answers:**

**Correct answer:**

To begin, solve the given equation for . This will give you the slope-intercept form of the line.

Divide everything by :

Therefore, the slope of the line is .

Now, for a point , the point-slope form of a line is:

, where is the slope

For our point, this is:

Distribute and solve for :

### Example Question #5 : Parallel Lines

Which of the following is parallel to the line running through the points and ?

**Possible Answers:**

**Correct answer:**

To begin, it is necessary to find the slope of the line running through the two points. (A parallel line will have the same slope. Recall that the slope is:

Or, for two points and :

For our points this is:

Now, to solve for this problem, the easiest way is to solve each equation for the form . When you do this, the slope () will be very easy to calculate. The only option that reduces to the correct slope is

Notice what happens when you solve for :

This shows that the slope of this line is .

### Example Question #1 : How To Find The Equation Of A Parallel Line

There is a line defined by the equation below:

There is a second line that passes through the point and is parallel to the line given above. What is the equation of this second line?

**Possible Answers:**

**Correct answer:**

Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = *–*3x + 12

y = *–*(3/4)x + 3

slope = *–*3/4

We know that the second line will also have a slope of *–*3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = *–*3/4(1) + b

2 = *–*3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = *–*(3/4)x + 2.75

### Example Question #1 : How To Find The Equation Of Parallel Lines

What is the equation of a line that is parallel to and passes through ?

**Possible Answers:**

**Correct answer:**

To solve, we will need to find the slope of the line. We know that it is parallel to the line given by the equation, meaning that the two lines will have equal slopes. Find the slope of the given line by converting the equation to slope-intercept form.

The slope of the line will be . In slope intercept-form, we know that the line will be . Now we can use the given point to find the y-intercept.

The final equation for the line will be .

### Example Question #2 : How To Find The Equation Of Parallel Lines

What line is parallel to and passes through the point ?

**Possible Answers:**

**Correct answer:**

Start by converting the original equation to slop-intercept form.

The slope of this line is . A parallel line will have the same slope. Now that we know the slope of our new line, we can use slope-intercept form and the given point to solve for the y-intercept.

Plug the y-intercept into the slope-intercept equation to get the final answer.

### Example Question #1 : Coordinate Geometry

What is the equation of a line that is parallel to the line and includes the point ?

**Possible Answers:**

**Correct answer:**

The line parallel to must have a slope of , giving us the equation . To solve for *b*, we can substitute the values for *y* and *x*.

Therefore, the equation of the line is .

### Example Question #3 : How To Find The Equation Of A Parallel Line

What line is parallel to , and passes through the point ?

**Possible Answers:**

**Correct answer:**

Converting the given line to slope-intercept form we get the following equation:

For parallel lines, the slopes must be equal, so the slope of the new line must also be . We can plug the new slope and the given point into the slope-intercept form to solve for the y-intercept of the new line.

Use the y-intercept in the slope-intercept equation to find the final answer.

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