So in recent videos, we've learned how to eliminate the t parameter from parametric equations to get back to equations just involving x and y. Well, some questions will actually have you do the opposite. They will give you a rectangular equation, and they'll ask you to find parametric equations for it. In other words, they'll ask you to write an x of t or a y of t. This is also sometimes called parameterizing an equation. It's really just the opposite of eliminating the parameter. So, I'm going to walk you through a step-by-step process of how to do this because there are a couple of things you want to keep in mind as you're doing this. Let's just jump right in.

Remember how when we eliminated the parameter, the basic idea was that we would solve one of these equations for t, then plug it into the other one and get rid of the t variable. Now we're doing the opposite. I'm going to go from an equation that involves just y and x, and I'm going to end up with an x of t and a y of t. So the first thing that you have to do here is you're going to have to choose an expression for t. You're going to have to pick something, an expression for t, either involving x or y. So what you're going to do here is if this is your equation, the first thing I can do for choosing t is I could just choose t to be x itself. It sounds kind of silly because I'm just choosing one variable to be another, but what happens? Well, what I'm going to do here is once I've chosen my t, is I'm going to solve for x of t. If t is equal to x, that means that x of t is just t itself. Right? x of t is just t. All I've done is swap the equation. So now that I've gotten one of my parametric equations, x of t is t, how do I get the second one, the y of t? All you're going to do is you're just going to substitute the x of t expression into the original equation and then solve for your y of t. So in other words, if I pop this into my original equation and every time I see x, I just replace it with t, what I end up with for y is I just end up with y equals 4(t+1). So all I've done here in this equation is I ended up with the same exact y equation. I've just traded the x variable for a t. It seems kind of silly because I just pick one variable to be another. But, in fact, if you actually were to plot out a bunch of t values, you would get the same exact x and y pairs as this equation. They actually just describe the same exact line.

So it turns out that setting t equal to x is pretty much almost always going to work in your problems. Now most problems, what they'll have you do is they'll have you sort of they'll prevent you from choosing x of t equals t because they want you to be a little bit more creative. Now this only worked because I chose t to be x, but I also could have chosen t to be something else. So let's come up with a different set of parametric equations for this. So again, I'm going to have to choose t, and it's going to usually involve some expression of x. So if I look through this equation over here, one of the things I can do is I could set t to be everything that's inside of this parenthesis over here. So I can set t not to just be x, but x + 1. Again, I'm just going to solve my x expression for t. So all I'm going to do is just subtract t-1. And then if I pop this into the original expression, what I'll end up with is that y = 4(t-1+1). The -1 +1 will cancel and leave me with just 4t. So, again, this is another perfectly valid set of parametric equations. If you were to plot out a bunch of values, you'll get the same x y pairs as this equation and also your first set as well. So it just depends on what you chose t to be. There's an infinite number of possibilities, but, usually, you're gonna want to choose something really simple.

That's really all there is to it. There's one thing that I haven't mentioned yet for choosing t, which is that you're gonna want to choose a t that avoids domain restrictions. So, for example, one