Video transcript
We're asked what is the equation of line B? And they tell us that line A has an equation y is equal to 2x plus 11. And they say that the line B contains the point 6, negative 7. And they tell us lines A and B are perpendicular, so that means that slope of B must be negative inverse of slope of A. So what we'll do is figure out the slope of A, then take the negative inverse of it. Then we'll know the slope of B, then we can use this point right here to fill in the gaps and figure out B's y-intercept. So what's the slope of A? This is already in slope-intercept form. The slope of A is right there, it's the 2, mx plus b. So the slope here is equal to 2. So the slope of A is 2. What is the slope of B? So what is B's slope going to have to be? Well, it's perpendicular to A, so it's going to be the negative inverse of this. The inverse of two is 1/2. The negative inverse of that is negative 1/2. So B's slope is negative 1/2. So we know that B's equation has to be y is equal to its slope, m times x plus some y-intercept. We still don't know what the y-intercept of B is, but we can use this information to figure it out. We know that y is equal to negative 7 when x is equal to 6. Negative 1/2 times 6 plus b, right? I just know that this is on the point, so this point must satisfy the equation of line B. So let's work out what b must be-- or what b, the y-intercept, this is a lowercase b, not the line B. So we have negative 7 is equal to-- what's negative 1/2 half times 6? That's not a b there, that's a 6. What's negative 1/2 times 6? It's negative 3, is equal to negative 3 plus our y-intercept. Let's add 3 to both sides of this equation, so if we add 3 to both sides-- I just want to get rid of this 3 right here-- what do we get? The left-hand side, negative 7 plus 3 is negative 4, and that's going to be equal to-- these guys cancel out-- that's equal to b, our y-intercept. So this right here is a negative 4. So the equation of line B is y is equal to-- its slope is a negative inverse of this character-- so negative 1/2, negative 1/2 x. And its y-intercept we just figured out is negative 4. And we are done.
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Find parallel and perpendicular lines step by step
The calculator will find the equation of the parallel/perpendicular line to the given line passing through the given point, with steps shown.
For drawing lines, use the graphing calculator.
Solution
Your input: find the equation of the line parallel to the line $$$y=2 x + 5$$$ passing through the point $$$\left(-3,5\right)$$$.
The equation of the line in the slope-intercept form is $$$y=2 x + 5$$$.
The slope of the parallel line is the same: $$$m=2$$$.
So, the equation of the parallel line is $$$y=2 x+a$$$.
To find $$$a$$$, we use the fact that the line should pass through the given point: $$$5=\left(2\right) \cdot \left(-3\right)+a$$$.
Thus, $$$a=11$$$.
Therefore, the equation of the line is $$$y=2 x + 11$$$.
Answer: $$$y=2 x + 11$$$.
#"given the slope m of a line then the slope of a line"#
#"perpendicular to it is"##•color(white)(x)m_(color(red)"perpendicular")=-1/m#
#"the equation of a line in "color(blue)"slope-intercept form"# is.
#•color(white)(x)y=mx+b#
#"where m is the slope and b the y-intercept"#
#"rearrange "5x-3y=2" into this form"#
#rArry=5/3x-2/3larr" with "m=5/3#
#rArrm_(color(red)"perpendicular")=-1/(5/3)=-3/5#
#rArry=-3/5x+blarrcolor(blue)"is the partial equation"#
#"to find b substitute "(-6,-1)#
#"into the partial equation"##-1=18/5+brArrb=-23/5#
#rArry=-3/5x-23/5larrcolor(red)"in slope-intercept form"#
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!
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:
Example: is y = 3x + 2 parallel to y − 2 = 3x ?
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)