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Write each pair of equations as a single equation in $x$ and $y$.
a)$\begin{cases} x=t+1 &\\ y=t^2-t & \\ \end{cases}$ b)$ \begin{cases} x=\sqrt[3]{t}-1 &\\ y=t^2-t & \\ \end{cases}$ c)$\begin{cases} x=\sin t &\\ y=\cos t & \\ \end{cases}$

All I want to know is what the question is asking me to do. Please do not give me the answer to any of these, if needed please make up an example. After that, I will edit with my steps to see if I am doing this correctly.

Edit: Now, I know this has come up before, but can someone please tell me the difference between $\arcsin$ and $\sin^{-1}$. Or are they the same?

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    What the question is asking you to do is this: Right now each of (a) (b) and (c) express x and y as functions of t. The problem would like you to just have one expression relating x and y. One way to do this is to solve the x or y expression for t, and then substitute into the other. Example: $x = t + 1$, $y = t - 1$. Then from the first equation $t = x - 1$, so substituting into the second $y = (x - 1) - 1 $, so $y = x-2$.2012-08-07
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    @AustinBroussard The method is the same, but the solution differs, because Jason solved a different problem ;)2012-08-08
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    @MTurgeon oh goodness. I was reading a different problem!2012-08-08
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    "...can someone please tell me the difference between..." - see [this](http://math.stackexchange.com/questions/30317)2012-08-08
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    @AustinBroussard Welcome to math.SE! I see that you have posted quite a few questions here. In order to help you, I would like to offer a couple of suggestions. First, if you don't understand what a homework exercise is asking, you should look at the examples in the chapter. Also, you should look at your notes from class for similar problems. These can help you get started. Finally, when you ask questions here and there is something you don't understand about the problem, we can more easily answer your question if you tell us which words in the exercise confuse you. Good luck with your HW!2012-08-08
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    I will also add that if you are already doing the things that I suggested, then post the examples you looked at or anything else you have done to try to solve the problem.2012-08-08

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$$(a)\,\,\,t=x-1\Longrightarrow y=t^2-t=(x-1)^2-(x-1)=(x-1)(x-2)\Longrightarrow y=(x-1)(x-2)$$ $$(b)\,\,\,x=\sqrt[3] t-1\Longrightarrow t=(x+1)^3....etc.$$ Can you now continue by yourself?

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    Is the next part of $(b)$, $=(x+1)^6-(x+1)^3$?2012-08-07
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    Indeed, and that equals $\,(x+1)^3\left((x+1)^3-1\right)\,$, if you love to factor stuff. For the last one think of Pythagoras Theorem... the trigonometric version.2012-08-07
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    I have to factor $(x+1)(x+1)(x+1)\Big((x+1)(x+1)(x+1)-1\Big)$???2012-08-08
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    Oh, dear god: not at all, unless threatened with a bazooka or something like that! Why would you do such a thing? Leave it as it is *unless specifically asked otherwise*2012-08-08
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    That's why I asked! So what do I do after $\,(x+1)^3\left((x+1)^3-1\right)\,$?2012-08-08
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    Go to a party, have a beer (only if you're over 21 in USA!), take a nap...or , if you want, *only* open up the inner parentheses: $$(x+1)^3-1=x^3+3x^2+3x=x(x^2+3x+3)$$and you can then, if you want, factor out that $\,x\,$ there...I'd leave as it is as that already solves what you asked.2012-08-08
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    Haha; so $y=x^3+3x^2+3x$ would be my final answer for $(b)$?2012-08-08
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    No @AustinBroussard, why?! We already had $$y=(x+1)^3\left[(x+1)^3-1\right]=(x+1)^3(x(x^2+3x+3))=x(x+1)^3(x^2+3x+3)$$Please do pay due attention to degree of polynomials: if you had degree $\,6\,$ at some point, then *unless there was cancelation* the final result *still must* have degree $\,6\,$2012-08-08
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For (c), think about a very familiar trig identity.

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Sorry, I posted the answers before without fully reading your question. Anyways, this type of question is called parametric. What you want to do is solve for t for one equation and then substitute that into the other equation. For the first one for example, t = x - 1 in the first equation. Now plug that t into the second equation.

Exact same thing for b (remember you want to get t ALONE)

For c, a hint is this: cos(arcsin x) or sin(arccos x) is always sqrt(1-x^2)

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    Your solution for (c) is lacking: $\,\sqrt{1-x^2}\,$ only gives the upper semicircle of the unit circle centered at the origin. You still need to express the lower semicircle.2012-08-08
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    Yes you are right. I realized that after posting that too, but I deleted my answer anyways. Thanks.2012-08-08