## Assignment # 10

**Parametric equations of a straight
line and their rectangular counterparts.**

We are investigating the following equation:

To begin with, we will let the domain for our t's range from
-2 pi ... 2pi. Our graph looks like this:

We can derive the rectangular equation from this based on the
slope and y-intercept, and get the following:

How do we arrive at that though?

We know that y = 2t -1. We also know
that x = t + 1, so t = x - 1. Now we substitute in the y equation
and we get y = 2 (x - 1) - 1 = 2x - 3.

So parametrics can be changed to rectangular coordinates by
a matter of solving for t and substituting. How does changing
a constant a little effect our overall graph? Let's look at the
generic equation:

Let's solve. Since x = at + b, we derive:

Substitute this in our y-equation:

After a little simplification, we get the following:

So now let's examine problem # 7 ...

**Problem # 7**

*" Write parametric equations of a line through (7,
5) with slope of 3. Graph the line using your equations."*

First, we put that equation in rectangular form:

So for the sake of simplicity, let's make a = 1 and c = 3.
Now we have the following:

Where 3b + d has to be 16. Let's make b = 5 and d = 1. This
gives us the following final setup:

On the other hand, we can have infinitely many equations. For
example, if a = 2, the only thing that has to change is c:

We can see that relationship ... On the other hand, if we were
to vary our b, it would effect only our d. Let's make b = 3:

All those are acceptable solutions to our given problem. Let's
refresh our setup that we are working with:

Generalizing this process to ANY line written in the form y
= mx + b (which we can find given a point and slope), we get the
following conditions:

And then it's just a matter of plugging the numbers in.

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