What is the slope of a line that is perpendicular to the line represented by the equation y 4x 1?

How to use Algebra to find parallel and perpendicular lines.

Parallel Lines

How do we know when two lines are parallel?

Their slopes are the same!

The slope is the value m in the equation of a line: y = mx + b

Example:

Find the equation of the line that is:

• parallel to y = 2x + 1
• and passes though the point (5,4)

The slope of y=2x+1 is: 2

The parallel line needs to have the same slope of 2.

We can solve it using the "point-slope" equation of a line:

y − y1 = 2(x − x1)

And then put in the point (5,4):

y − 4 = 2(x − 5)

And that answer is OK, but let's also put it in y = mx + b form:

y − 4 = 2x − 10

y = 2x − 6

Vertical Lines

But this does not work for vertical lines ... I explain why at the end.

Not The Same Line

Be careful! They may be the same line (but with a different equation), and so are not parallel.

How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope:

For y = 3x + 2: the slope is 3, and y-intercept is 2

For y − 2 = 3x: the slope is 3, and y-intercept is 2

In fact they are the same line and so are not parallel

Perpendicular Lines

Two lines are Perpendicular when they meet at a right angle (90°).

To find a perpendicular slope:

When one line has a slope of m, a perpendicular line has a slope of −1m

In other words the negative reciprocal

Example:

Find the equation of the line that is

• perpendicular to y = −4x + 10
• and passes though the point (7,2)

The slope of y=−4x+10 is: −4

The negative reciprocal of that slope is:

m = −1−4 = 14

So the perpendicular line will have a slope of 1/4:

y − y1 = (1/4)(x − x1)

And now put in the point (7,2):

y − 2 = (1/4)(x − 7)

And that answer is OK, but let's also put it in "y=mx+b" form:

y − 2 = x/4 − 7/4

y = x/4 + 1/4

Quick Check of Perpendicular

When we multiply a slope m by its perpendicular slope −1m we get simply −1.

So to quickly check if two lines are perpendicular:

When we multiply their slopes, we get −1

Like this:

Are these two lines perpendicular?

When we multiply the two slopes we get:

2 × (−0.5) = −1

Yes, we got −1, so they are perpendicular.

Vertical Lines

The previous methods work nicely except for a vertical line:

In this case the gradient is undefined (as we cannot divide by 0):

m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined

So just rely on the fact that:

• a vertical line is parallel to another vertical line.
• a vertical line is perpendicular to a horizontal line (and vice versa).

Summary

• parallel lines: same slope
• perpendicular lines: negative reciprocal slope (−1/m)

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What is the slope of any line perpendicular to 2y = 4x +3 ?

Possible Answers:

Explanation:

First, we must solve the equation for y to determine the slope:  y = 2x + 3/2

By looking at the coefficient in front of x, we know that the slope of this line has a value of 2. To fine the slope of any line perpendicular to this one, we take the negative reciprocal of it:

slope = m , perpendicular slope = – 1/m

slope = 2 , perpendicular slope = – 1/2

What line is perpendicular to 2x + y = 3 at (1,1)? 

Possible Answers:

Correct answer:

x – 2y = -1

Explanation:

Find the slope of the given line.  The perpendicular slope will be the opposite reciprocal of the original slope.  Use the slope-intercept form (y = mx + b) and substitute in the given point and the new slope to find the intercept, b.  Convert back to standard form of an equation:  ax + by =  c.

What is the slope of the line perpendicular to the line given by the equation

6x – 9y +14 = 0

Possible Answers:

Explanation:

First rearrange the equation so that it is in slope-intercept form, resulting in y=2/3 x + 14/9. The slope of this line is 2/3, so the slope of the line perpendicular will have the opposite reciprocal as a slope, which is -3/2.

What is the slope of the line perpendicular to the line represented by the equation y = -2x+3?

Possible Answers:

Explanation:

Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2.

Find the slope of a line perpendicular to the line y = –3x – 4.

Possible Answers:

Explanation:

First we must find the slope of the given line. The slope of y = –3x – 4 is –3. The slope of the perpendicular line is the negative reciprocal. This means you change the sign of the slope to its opposite: in this case to 3. Then find the reciprocal by switching the denominator and numerator to get 1/3; therefore the slope of the perpendicular line is 1/3.

What is the slope of a line perpendicular to the following:

Possible Answers:

Correct answer:

Explanation:

The question puts the line in point-slope form y – y1 = m(x – x1), where m is the slope. Therefore, the slope of the original line is 1/2.  A line perpendicular to another has a slope that is the negative reciprocal of the slope of the other line. The negative reciprocal of the original line is –2, and is thus the slope of its perpendicular line. 

A line is defined by the following equation:

What is the slope of a line that is perpendicular to the line above?

Possible Answers:

Correct answer:

Explanation:

The equation of a line is 

Rearrange the equation to match this:

For the perpendicular line, the slope is the negative reciprocal;

therefore 

Which of the following lines is perpendicular to the line passing through the points 

Possible Answers:

Correct answer:

Explanation:

First, you must find the slope of the line given to you. Remember that the slope is calculated: 

Thus, for our data, this is:

Now, the perpendicular slope to this is opposite and reciprocal. Hence, it must be 

To know this, solve the equation for the format 

Next, divide everything by 

You really just need to pay attention to the 

What is the slope of a line that is perpendicular to the equation given by:

Possible Answers:

Correct answer:

Explanation:

Perependicular lines have slopes whose product is 

What is the slope of a line perpendicular to line 

Possible Answers:

Correct answer:

Explanation:

To find the slope of a perpendicular line, we take the reciprocal 

The easy way to do this is to simply take the fraction (a whole slope can be made into a fraction by placing 

However, if we attempt to follow this procedure, we get:

Thus, our perpendicular line (which is a vertical line) has an undefined slope.

Reuben
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Florida State University, Bachelor of Science, Environmental Science.

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