Chapter 4.3, Problem 28E

To determine

To find: an equation that represents the new path.

Answer to Problem 28E

Therefore, the equation of the new path is $y=\frac{4}{3}x-\frac{1}{3}$

Explanation of Solution

Given:

Consider the line passing through the points $\left(8,0\right)$ and $\left(11,4\right)$ .

The objective is to find the equation that represents the path which is parallel to the line which passes through the points $\left(8,0\right)$ and $\left(11,4\right)$ .

Calculation:

The slope of line which passing through two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is $m=\frac{y_2-y_1}{x_2-x_1}$

Therefore, the slope of the line $\left(8,0\right)$ and $\left(11,4\right)$

  $m=\frac{4-0}{11-8}$

  $=\frac{4}{3}$

Two lines are parallel if their slopes are equal.

The other line passing through the point $\left(4,5\right)$ has the same slope.

Use the formula $y-y_1=m\left(x-x_1\right)$ to find the equation of the line passing through $\left(4,5\right)$ and

Slope $m=\frac{4}{3}$

This implies,

  $y-y_1=m\left(x-x_1\right)$

  $y-5=\frac{4}{3}\left(x-4\right)$

  $\text{y=}\frac{4}{3}x-\frac{1}{3}$

Hence, the equation of the new path $\text{y=}\frac{4}{3}x-\frac{1}{3}$

Conclusion:

Therefore, the equation of the new path is $y=\frac{4}{3}x-\frac{1}{3}$