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)
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There is a line defined by the equation below:
There is a second line that passes through the point
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
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
The final equation for the line will be
What line is parallel to
Possible Answers:
Correct answer:
Explanation:
Start by converting the original equation to slop-intercept form.
The slope of this line is
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
Possible Answers:
Correct answer:
Explanation:
The line parallel to
Therefore, the equation of the line is
What line is parallel to
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
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:
Parallel lines have the same slope, so now we need to find the equation of a line with slope
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 "
So the slope of the original line is
So, the slope is
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
Take your equation:
First, subract
Next, subtract
Finally, divide by
Thus, your slope is
Among the options provided only
First, subtract
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
Therefore we want to find an equation that has the same
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
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