Chapter 10.4, Problem 20E

To determine

To draw: The graph of the line if the equations are $x=5-2t$ , $y=1+t$ , and $z=5-\frac{1}{2}t$ .

Explanation of Solution

Given information:

The equations are $x=5-2t$ , $y=1+t$ , and $z=5-\frac{1}{2}t$ .

Graph:

The graph of the line with equation $x=5-2t$ , $y=1+t$ , and $z=5-\frac{1}{2}t$ is shown below.

  

Interpretation:

To draw a line, two points are required. The two points can be calculated by considering two different value of $t$ .

Assume the value of $t$ as 2. Substitute the value of $t$ as 2 in the given equations.

  $\begin{array}{c}x=5-2\left(2\right)\\ =5-4\\ =1\end{array}$

and,

  $\begin{array}{c}y=1+2\\ =3\end{array}$

and,

  $\begin{array}{c}z=5-\frac{1}{2}\left(2\right)\\ =5-1\\ =4\end{array}$

From the above, the coordinate of the point is $\left(1,3,4\right)$ when the value of $t$ is 2.

Assume the value of $t$ as 4. Substitute the value of $t$ as 4 in the given equations.

  $\begin{array}{c}x=5-2\left(4\right)\\ =5-8\\ =-3\end{array}$

and,

  $\begin{array}{c}y=1+4\\ =5\end{array}$

and,

  $\begin{array}{c}z=5-\frac{1}{2}\left(4\right)\\ =5-2\\ =3\end{array}$

From the above, the coordinate of the point is $\left(-3,5,3\right)$ when the value of $t$ is 4.

Draw the graph of the line that passes through the point $\left(1,3,4\right)$ and $\left(-3,5,3\right)$ .