Answer :
Since the path of Grant is a line segment, the first step to solve the exercise is to find the slope of the line segment. For this, we can use the following formula:
[tex]\begin{gathered} m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \text{ Where m is the slope of the line, and} \\ (x_1,y_1)\text{ and }(x_2,y_2)\text{ are two points through the line passes} \end{gathered}[/tex]As we can see in the graph, the line segment passes through the points (8,0) and (-4,6). Then, we have:
[tex]\begin{gathered} (x_1,y_1)=(8,0) \\ (x_2,y_2)=(-4,6) \\ m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m=\frac{6-0}{-4-8} \\ m=\frac{6}{-12} \\ \text{ Simplify} \\ m=\frac{1\cdot6}{-2\cdot6} \\ m=-\frac{1}{2} \end{gathered}[/tex]Now that we have the slope of the line segment and a point which it passes, we can use the point-slope formula:
[tex]y-y_1=m(x-x_1)\Rightarrow\text{ Point-slope formula}[/tex][tex]y-0=-\frac{1}{2}(x-8)[/tex]Finally, we solve for y the above equation:
[tex]\begin{gathered} y=-\frac{1}{2}(x-8) \\ \text{ Apply the distributive property} \\ y=-\frac{1}{2}\cdot x-\frac{1}{2}\cdot-8 \\ y=-\frac{x}{2}+\frac{1}{2}\cdot8 \\ y=-\frac{x}{2}+\frac{8}{2} \\ y=-\frac{x}{2}+4 \\ \text{ Reorder} \\ y=4-\frac{x}{2} \end{gathered}[/tex]Therefore, the equation that represents the path of Grant is:
[tex]$\boldsymbol{y=4-\frac{x}{2}}$[/tex]