How to find the equation of a parallel line without points

Table of Contents Show

Parallel Lines
Vertical Lines
Not The Same Line
Perpendicular Lines
Quick Check of Perpendicular
Vertical Lines

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Lessons

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For example, if I need to find the equation of the line parallel to $$2x-3y=4$$ which passes through the point $(1,-5)$ I know how to do this by putting it into slope-intercept form first to find the slope and then plugging in the point to find the y-intercept.

Same thing for finding a line perpendicular to that point. I just wanted to know if I could do this without changing it into slope intercept form first

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?

Line	Slope
y = 2x + 1	2
y = −0.5x + 4	−0.5

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)

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:

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

What is the equation of a line that is parallel to

and passes through

Possible Answers:

Correct answer:

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

What line is parallel to

and passes through the point

Possible Answers:

Correct answer:

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.

What is the equation of a line that is parallel to the line

and includes the point

Possible Answers:

Correct answer:

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

What line is parallel to

, and passes through the point

Possible Answers:

Correct answer:

Explanation:

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.

What line is parallel to

Possible Answers:

None of the answers are correct

Correct answer:

Explanation:

Find the slope of the given line:

(slope intercept form)

therefore the slope is

Parallel lines have the same slope, so now we need to find the equation of a line with slope

and going through point

by substituting values into the point-slope formula.

So,

Thus, the new equation is

Which of these formulas could be a formula for a line perpendicular to the line

Possible Answers:

Correct answer:

Explanation:

This is a two-step problem. First, the slope of the original line needs to be found. The slope will be represented by "

" when the line is in

-intercept form

So the slope of the original line is

. A line with perpendicular slope will have a slope that is the inverse reciprocal of the original. So in this case, the slope would be

. The second step is finding which line will give you that slope. For the correct answer, we find the following:

So, the slope is

, and this line is perpendicular to the original.

Which of the following is a line that is parallel to the line defined by the equation

Possible Answers:

Correct answer:

Explanation:

Since parallel lines have equal slopes, you should find the slope of the line given to you. The easiest way to do this is to solve the equation so that its form is

represents the slope.

Take your equation:

First, subract

from both sides:

Next, subtract

from both sides:

Finally, divide by

, which is the same as

Thus, your slope is

Among the options provided only

is parallel. Solve this equation as well for

form.

First, subtract

from both sides:

Then, divide by

Which of the following answer choices gives the equation of a line parallel to the line:

Possible Answers:

Correct answer:

Explanation:

Parallel lines have the same slope but different y-intercepts. When the equations of two lines are the same they have infinitely many points in common, whereas parallel lines have no points in common.

Our equation is given in slope-intercept form,

where

is the slope. In this particular situation

Therefore we want to find an equation that has the same

value and a different

value.

Thus,

is parallel to our equation.

What is the equation of a line parallel to the line given by the equation:

Possible Answers:

Correct answer:

Explanation:

Parallel lines have the same slope and differing y-intercepts. Since

is the only equation with the same slope, and the y-intercept is different, this is the equation of the parallel line.

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