Chapter 8.4, Problem 29E

To determine

To Determine: The parametric equation for the line

Answer to Problem 29E

The slope- intercept form is $y=\frac{x}{2}-3$ .

Explanation of Solution

Given:

The Slope $\frac{1}{2}$ , passing through $(4,-1)$

Calculation:

Find the parametric equation for the line with the Slope $\frac{1}{2}$ , passing through $(4,-1)$ .

start at the point $(4,-1)$ and move up and to the right along the line. Since the line has a slope $=\frac{1}{2}$ , for every 1 unit move to the right, move up $\frac{1}{2}$ units.

Therefore, the increase in the x- coordinates by t units leads to increase in the y-coordinates by $\frac{t}{2}$ units.

Thus, the parametric equation for the line with the Slope $\frac{1}{2}$ , passing through $(4,-1)$ are $\boxed{x=4+t}$ and $\boxed{y=-1+\frac{1}{2}t}$ .

To confirm that these equations give the line of slope $\frac{1}{2}$ and passes through the points $(4,-1)$ .

Eliminate the parameter t.

Therefore, on putting $t=x-4$ in $y=1+\frac{1}{2}t$

  $\begin{array}{l}y=1+\frac{1}{2}(x-4)\\ y=\frac{-2+x-4}{2}\\ y=\frac{x-6}{2}\\ y=\frac{x}{2}-3\end{array}$

Thus, the slope- intercept form of the equation of this line of slope $\frac{1}{2}$ and passes through the point $(4,-1)$ is $y=\frac{x}{2}-3$ .the graph of the equation $y=\frac{x}{2}-3$ is

  

Conclusion:

Therefore, the slope- intercept form of the equation of this line of slope $\frac{1}{2}$ and passes through the point $(4,-1)$ is $y=\frac{x}{2}-3$ .