About this video
The video writes equations of lines parallel to a given line through a given point. Parallel lines have the same slope, so the line parallel to y = -2/3 x - 1 through (9, 2) is y = -2/3 x + 8. It also covers special cases: a line parallel to y = 4 through (-7, -3) is y = -3, and a line parallel to x = -8 through (3, -2) is x = 3.
You'll be able to
- use the fact that parallel lines have the same slope.
- substitute a point into y = mx + b to find b.
- write equations of horizontal (y = c) and vertical (x = c) lines parallel to a given line through a point.
Key words
- parallel lines
- Lines that never meet; non-vertical parallel lines have the same slope.
- slope
- How steep a line is; the m in y = mx + b.
- y-intercept
- Where the line crosses the y-axis; the b in y = mx + b.
- horizontal line
- A flat line with slope 0, written y = a number.
- vertical line
- A straight up-and-down line, written x = a number; its slope is undefined.
Transcript
- [Instructor] What we're going to do in this video is get some practice writing equations of parallel lines. So they say here, "Write the equation of a line that is parallel to y is equal to negative 2/3 x minus one and that passes through the 9,2. So pause this video, have a go at this before we do this together.
Okay, so the first thing that we know about a parallel line is that it's going to have the exact same slope as our original line. So our line is going to have a slope of negative 2/3. So we know that our line is going to have the form y is equal to negative 2/3 x.
Now what we don't know is exactly what's the y intercept going to be, so I'll just put, say, plus b right over year. Now this is, we're making progress, but this can't be the equation 'cause we need to know what that y intercept is, but luckily they have told us a point that it goes through.
So what we can do is is we can substitute the nine for x, the two for y, and then solve for b. So let's do that. So we know that when x is nine, y is two. So two is equal to negative 2/3 times nine plus b. And let's see. If we just do this over here, I could say that two is equal to negative 2/3 times nine.
2/3 times nine is six, so this would be negative six plus b. To solve for b, I just add six to both sides, and I get eight is equal to b. Now we're not fully done. I want to write the whole equation, so let me just rewrite this, but instead of b, I'm now going to write eight.
So y is equal to negative 2/3 x plus eight. And we're done. We've done that one. Let's do a couple of other examples, and these in some ways are special cases, but as we'll see, the math is actually more straightforward for some of these, or actually for both of these.
So here we're told, "Write the equation of line that is parallel to y equals four and that passes through the point -7,-3. Pause this video. See if you can do that. So some of you might say, "Hey, this is a little bit different. What's the slope here? I don't even see an x term."
And I can give you a hint. This is the same thing as saying y is equal to zero x plus four. Now when you write it in that form, it becomes very clear that the slope is zero. Or if you were to graph this, this is just a horizontal line, so it's going to have a slope of zero.
So a parallel line to that is also going to have a slope at zero. It's just going to be a different constant y value. And if it passes through this point, -7,-3. its y value needs to go through, or it needs to intersect negative three. And since it has a slope of zero, it's always going to be y equals negative three.
So that's actually our answer. I can write it over here. And the second one, maybe I'll do it in blue, y is equal to negative three. Remember, the logic here is we want just a horizontal line. Horizontal lines are just y equals something. Any horizontal line is parallel to any other horizontal line.
And so we just essentially are shifting where our y is. So y is going to be equal to negative three everywhere for this second line, but it's gonna be parallel to y equals four. Now let's do another case that in some ways is the opposite but very similar. "Write the equation of a line that is parallel to x equals negative eight and that passes through the point 3,-2."
Pause the video. Have a go at that. All right, so some of you might recognize if x is always equal to negative eight, this is actually a vertical line. And it's actually hard to write it in traditional slope intercept form because you actually have an infinite slope in this situation, or you could even think of it as a negative infinite slope, depending on how you want to think about it.
But if you just think about, all right, if x is equal... Let me do this in another color. So let me do it in orange. If x is equal to negative eight, any other vertical line that just has a constant x is going to be parallel to that. And so we need a vertical line that has a constant x that goes through this point.
Well, if it has a constant x, it must go through x equaling three, and it has to be the same thing the whole time. It's a vertical line. So x is equal to three is our answer.
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