Chapter 9.4, Problem 16E

(a)

To determine

To calculate: To sketch the curve represented by the parametric equations

Explanation of Solution

Given information: Parametric equations $x=3-2t;y=2+3t$

Calculation:

Parametric equations are given as

  $x=3-2t;y=2+3t$

Calculating the values of $x$ and $y$ at various values of $t$

$t$	$x$	$y$
-3	9	-7
-2	7	-4
-1	5	-1
0	3	2
1	1	5
2	-1	8
3	-3	11

Graph of the function is as follows:

The arrows on the curve indicates its orientation as $t$ increases from $-3$ to $3$

  

Conclusion:

Hence, graph is sketched

(b)

To determine

To calculate: To write the rectangular equations for the parametric equations

Answer to Problem 16E

The rectangular equation is $3x+2y-13=0$

Explanation of Solution

Given information: Parametric equations $x=3-2t;y=2+3t$

Calculation:

Parametric equations are given as

  $x=3-2t;y=2+3t$

From equation $x=3-2t$

  $\begin{array}{l}x=3-2t\\ t=\frac{3-x}{2}----(1)\end{array}$

Substitute $t=\frac{x-3}{2}$ in $y=2+3t$ ,

  $\begin{array}{l}y=2+3\left(\frac{3-x}{2}\right)\\ y=2+\frac{9-3x}{2}\\ y=\frac{4+9-3x}{2}\\ 2y=13-3x\\ 3x+2y-13=0\end{array}$

Here, domain is $-\infty <t<\infty$

Conclusion:

Hence, rectangular equation is $3x+2y-13=0$