# GRE Math : How to find the equation of parallel lines

## Example Questions

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

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

y = -11/6x + 8.32

y = 6/11x - 33

y = 15x + 12

y = 11/6x + 88

y = -6/11x + 57.4

y = -6/11x + 57.4

Explanation:

(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 #1 : Coordinate Geometry

Lines m and n are parallel

What is the value of angle ?

125

180

145

115

130

145

Explanation:

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 #1 : Coordinate Geometry

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

Explanation:

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 #4 : Coordinate Geometry

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

Explanation:

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 #4 : Coordinate Geometry

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

Explanation:

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 : Parallel Lines

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?

Explanation:

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 : Parallel Lines

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

Explanation:

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 #1 : Parallel Lines

What line is parallel to   and passes through the point ?

Explanation:

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 #2 : Lines

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

Explanation:

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 #2 : How To Find The Equation Of A Parallel Line

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