Converting parametric equations to xy equation I have these parametric equations:
$$
x=1+t \\
y=1-t
$$
How do I make this to the xy equation?


*

*David

 A: *

*Use the first equation to find an expression for $t$ in terms of $x$.

*Substitute into the second equation to get an equation only involving $x$ and $y$.

*Rearrange if desired.  



Update: This method will work for the example you gave, but might not work for a general example.  For example, the following is a particularly common parametrization: 
\begin{align}
x &= \cos t \\
y &= \sin t
\end{align}
Here, you can't get a unique expression for $t$ in terms of $x$: for example, if $x$ is $1$, we could have $t=0$ or $t=2\pi$.  In this case, the solution is to square both equations to give:
\begin{align}
x^2&=\cos^2 t\\
y^2&=\sin^2 t
\end{align}
Then add them together to get
$$
x^2+y^2=\cos^2t+\sin^2t=1
$$
This equation ($x^2+y^2=1$) is the equation for a circle of radius $1$ about $(0,0)$.  

Edit: (answering the question in comment below) Sometimes you're presented with a parametrization like
\begin{align}
x&=1+t\\
y&=1-t
\end{align}
and you want to convert that into a vector equation.  A mathematician would immediately recognize that system of parametric equations as a vector equation: $x$ and $y$ are themselves coordinates of the vector $\begin{pmatrix}
x\\ 
y
\end{pmatrix}$.  The equations then become: 
$$
\vec{r}=\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}1+t\\1-t\end{pmatrix}
$$
and you can rearrange that into whatever (vector) form you like using the usual rules for addition of vectors.  
A: $\text{AD}$$\text{D}$ $\text{EM}$.
A: $x = 1 + t
\\y = 1 - t 
\\x = -(1- t) + 2 
\\x = 2 - y  $
