Equation of a Line Using Point-Slope Formula

Example No 1

Let's do an example. Let's say that at 1 p.m., I start out at mile marker 10, and I'm going to go 72 mph. Let's use point-slope form to relate the time with my location. Point-slope form isy=y sub 1 + m(x - x sub 1). Let's start filling things in. m is the slope, so that's 72, because I'm going 72 mph. x is x; that's my time. x sub 1 and y sub 1 are my current location. So x sub 1 is the current time (this point here), which is just 1 for 1 p.m. y sub 1 is the location at 1 p.m., which is just mile marker 10. I can plug in those two numbers to get y=10 + 72(x - 1). If I want to know where I am at any given point in time, I can plug in that value for x and solve to find my location, y. So, at 4 p.m., x=4 and y is then 10 + 72(4 - 1), which is 226 miles. That's my new mile marker.

Example No 2

All right, let's go back to this wedding problem. Let's say I'm not driving. After going 72 mph, I got too many speeding tickets, so I'm letting someone else do the driving. The wedding is at 10 p.m., and it's at mile marker 100 in Las Vegas. Currently, it's 2 p.m. and we're at mile marker 5. I leave it up to my buddy and I fall asleep in the back seat. At 6 p.m., I wake up and we're at mile marker 25. Is there any way that we're going to get to Vegas by 10 p.m.? Here I don't know how fast we're going, so I can't use point-slope formula. But I do have two points, and I know that we're going a constant speed. I can find out what our velocity has been between 2 p.m. and now and use that to find my speed for my point-slope formula. Let's first find the slope. The two points are (2, 5), because at 2 p.m. I was at mile marker 5, and (6, 25) because at 6 p.m. I was at mile marker 25. I can find the slope of this line: delta y / delta x, which is the change in location divided by the change in time. My location has gone from 5 to 25, which is a difference of 20, and time has gone from 2 to 6, which is 4 hours. So, I've gone 20 miles in 4 hours. That gives me 5 mph. I really need to talk to this guy.

We're on the freeway here. Why are we going to 5 mph? Is there really bad traffic or something? That's neither here nor there. I know my slope now, and I have a point. In fact, I have two points. And I can use either one in the point-slope form to find out an equation for where I will be at what time.

So point-slope form says that y=y sub 1 + m(x - x sub 1). I'm going to use my current location as my point, so y sub 1 is 25, for mile marker 25, x sub 1 is 6, because it's 6 p.m., and m is my speed, which is a lousy 5 mph. Point-slope form gives me y=25 + 5(x - 6). I can simplify this and I end up with y=5x - 5. Where will I be at 10 p.m.? At 10 p.m., x will be 10. Plugin x=10, and I get 50 - 5, which is y=45. I will be at mile marker 45. That's nowhere close to Vegas. Vegas is at mile marker 100. When will I get to Las Vegas? This means that I need to find what value of x will give me a value of y that's 100. Let's solve 100=5x - 5. I'm going to add 5 to both sides, then divide by 5, and find that x=21 hours after I started. That means I won't get there until 9 a.m. tomorrow. Well, so much for celebrating that wedding. I probably won't even make the after-party.

Lesson Summary

Let's recap. To find an equation for the line that goes through the point (x sub 1, y sub 1) with some slope m, we use the point-slope formula: y=y sub 1 + m(x - x sub 1). We can use this, for example, to find out where we're going to be at any given point in time on our road trip to Las Vegas.