What is the equation of the line that is perpendicular to and that passes through the point write the equation in slope intercept form?

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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 line is perpendicular to x + 3y = 6 and travels through point (1,5)?

Possible Answers:

Correct answer:

y = 3x + 2

Explanation:

Convert the equation to slope intercept form to get y = –1/3x + 2.  The old slope is –1/3 and the new slope is 3.  Perpendicular slopes must be opposite reciprocals of each other:  m1 * m2 = –1

With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2

So y = 3x + 2

What line is perpendicular to

Possible Answers:

Correct answer:

Explanation:

Convert the given equation to slope-intercept form.

The slope of this line is

The perpendicular slope is 

Plug the new slope and the given point into the slope-intercept form to find the y-intercept.

So the equation of the perpendicular line is

What is the equation of a line that runs perpendicular to the line 2x + y = 5 and passes through the point (2,7)?

Possible Answers:

Correct answer:

–x/2 + y = 6

Explanation:

First, put the equation of the line given into slope-intercept form by solving for y. You get y = -2x +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2x + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½x + 6. Rearranged, it is –x/2 + y = 6.

Line m passes through the points (1, 4) and (5, 2). If line p is perpendicular to m, then which of the following could represent the equation for p?

Possible Answers:

Correct answer:

2x – y = 3

Explanation:

The slope of m is equal to   y2-y1/x2-x1  =  2-4/5-1 = -1/2                                  

Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be –1:

(slope of p) * (-1/2) = -1

Slope of p = 2

So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x – y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.

What is the equation for the line that is perpendicular to 

Possible Answers:

Correct answer:

Explanation:

Perpendicular slopes are opposite reciprocals.

The given slope is found by converting the equation to the slope-intercept form.

The slope of the given line is

We can use the given point and the new slope to find the perpendicular equation. Plug in the slope and the given coordinates to solve for the y-intercept.

Using this y-intercept in slope-intercept form, we get out final equation:

Which line below is perpendicular to

Possible Answers:

Correct answer:

Explanation:

The definition of a perpendicular line is one that has a negative, reciprocal slope to another.

For this particular problem, we must first manipulate our initial equation into a more easily recognizable and useful form: slope-intercept form or

According to our 

Our answer will have a slope of

(As an aside, the negative reciprocal of 4 is

If a line has an equation of 

Possible Answers:

Correct answer:

Explanation:

Putting the first equation in slope-intercept form yields

A perpendicular line has a slope that is the negative inverse. In this case,

Which of the following is possibly a line perpendicular to 

Possible Answers:

Correct answer:

Explanation:

To start, begin by dividing everything by 

Now, recall that the slope of a perpendicular line is the opposite and reciprocal slope to its mutually perpendicular line.  Thus, if our slope is 

Which of the following is the equation of a line perpendicular to the line given by:

Possible Answers:

Correct answer:

Explanation:

For two lines to be perpendicular their slopes must have a product of 

What is the equation of a line perpendicular to the line defined by the equaiton:

Possible Answers:

Correct answer:

Explanation:

Perpendicular lines have slopes whose product is 

Looking at our equations we can see that it is in slope-intercept form where the m value represents the slope of the line,

In our case we see that

 therefore, 

Since 

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