Writing equations of perpendicular lines
About this video
Line A is y = 2x + 11, and line B is perpendicular to it through the point (6, -7). The video takes the negative inverse of A's slope to get -1/2, then substitutes the point to find the y-intercept -4, giving line B: y = -1/2 x - 4.
You'll be able to
- find the slope of a line perpendicular to a given line.
- use a point on a line to find its y-intercept.
- write the equation of a perpendicular line in slope-intercept form.
Key words
- perpendicular lines
- Lines that meet at right angles; their slopes are negative inverses.
- negative inverse
- Flip a number and change its sign; the negative inverse of 2 is -1/2.
- y-intercept
- The value b in y = mx + b, where the line crosses the y-axis.
- slope-intercept form
- y = mx + b, with slope m and y-intercept b.
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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