Chapter 8.4, Problem 32E

To determine

The parametric equation for the line

Answer to Problem 32E

The slope −intercepts form is $y=\frac{7}{12}x$ .

Explanation of Solution

Given:

Point $(12,7)$ and the origin

Calculation:

Find the parametric equation, first find the slope.

The slope $m$ is given by

  $m=\frac{y_2-y_1}{x_2-x_1}$ , where $(x_1,y_1)=(12,7)$ and $(x_2,y_2)=(0,0)$ .

On substituting $x_1=12,x_2=7,y_1=0and{y}_2=0$ , we get

  $\begin{array}{l}m=\frac{0-7}{0-12}\\ m=\frac{7}{12}\end{array}$

Start at the point $(12,7)$ and move up and to the right along the line. Because the line has slope = $\frac{7}{12}$ , for every 1 unit we move to right , move up $\frac{7}{12}$ units.

Thus, the parametric equations for the line of slope $\frac{7}{12}$ and passes through the points $(12,7)$ and (0,0).

  $x=12+t$ and $y=7+\frac{7}{12}t$ .

To confirm that these equations give the line of slope $\frac{7}{12}$ and passes through the points $(12,7)$ eliminate the parameter t.

Therefore, on putting $t=x-12$ in $y=7+\frac{7}{12}t$ we get

  $\begin{array}{l}y=7+\frac{7}{12}(x-12)\\ y=\frac{7}{12}x\end{array}$

Thus, the slope −intercepts form of the equation of this line of slope $\frac{7}{12}$ and passes through the points $(12,7)$ and (0,0) is $y=\frac{7}{12}x$ .

The graph of the equation $y=\frac{7}{12}x$ is

  

Conclusion:

Therefore, the slope −intercepts form of the equation of this line of slope and passes through the points $(12,7)$ and (0,0) is $y=\frac{7}{12}x$ .