Question

In Exercises 24–34 sketch the parametric curve by eliminating the parameter

24.$x=2t-1,y=3t+5,t\in \mathrm{ℝ}$

Short answer

As a result, after removing the parameter, the needed equation is$\mathit{y}\mathbf{=}\mathbf{3}\mathbf{·}\frac{\mathbf{x}\mathbf{+}\mathbf{1}}{\mathbf{2}}\mathbf{+}\mathbf{5}$

Step-by-step solution

Given information

Given:$x=2t-1,y=3t+5,t\in \mathrm{ℝ}$

Removing the parameter

Take the parametric curve $x=2t-1$

$1$should be added to both sides of the equation:

$$\begin{gathered} x+1=2t-1+1 \\ \Rightarrow x+1=2t \end{gathered}$$

Divide both sides by $2$:

$$\begin{gathered} \frac{x+1}{2}=\frac{2t}{2} \\ \Rightarrow \frac{x+1}{2}=t \end{gathered}$$

Substitute the parametric equation $t=\frac{x=1}{2}$in place of both sides of the equation, $y=3t+5$

Then you've got :

$y=3·\frac{x+1}{2}+5$

As a result, after removing parameter $1$, the needed equation is $y=3·\frac{x+1}{2}+5$

Substituting different values for x

To create the equation's graph assume $x=-1,0,1,2$:

1) Substitute $x=-1$in the equation :

$$\begin{gathered} y=5 \\ \Rightarrow (x,y)=(-1,5) \end{gathered}$$

2) Substitute $x=0$:

$$\begin{gathered} y=1.5+5 \\ \Rightarrow y=6.5(0,6.5) \\ \Rightarrow (x,y)=(0,6.5) \end{gathered}$$

3) Replace $x=1$:

$$\begin{gathered} y=3(1+1)+5 \\ =8 \\ \Rightarrow (x,y)=(1,8) \end{gathered}$$

4) Substitute $x=2$:

$$\begin{gathered} y=3(2+1) \\ =4.5+5 \\ =9.5 \\ \Rightarrow (x,y)=(2,9.5) \end{gathered}$$

Plotting the graph

The following is the graphical representation using the points $(-1,5),(0,6.5),(1,8),(2,9.5)$:

Therefore, the required equation after eliminating the parameter is$y=3\frac{(x+1)}{2}+5$