In order for the equation to represent a line that is parallel to the line that is shown, the equation must have the same slope as line segment \(\displaystyle \bar{AB}\). 

Since, line segment \(\displaystyle \bar{AB}\) has a slope of \(\displaystyle \frac{9}{13}\), the correct equation is: \(\displaystyle y=\frac{9}{13}-8\)

The above line segment is a horizontal line that passes through the \(\displaystyle y\) axis at \(\displaystyle -7.\) Since this line is horizontal, it does not have a slope. Therefore, \(\displaystyle y=-7\) is the correct answer. 

Since the line segment is horizontal, the equation that is perpendicular to the segment must run vertically. The only linear equation that runs vertically (perpendicular to \(\displaystyle y=-7\)) is \(\displaystyle x=-3\).

Since, \(\displaystyle x=-3\) is perpendicular to \(\displaystyle y=-7\) the points must cross at \(\displaystyle (-3,-7)\), because it is the only coordinate point that both lines pass through. 

Keep in mind that the values in the coordinate points are \(\displaystyle (x,y)\), thus the point \(\displaystyle (3,0)\) is the point at which the line segment passes through the \(\displaystyle x\) axis. 

To identify the correct equation, apply the formula \(\displaystyle y=mx+b\), where \(\displaystyle m\) represents the slope of the line and \(\displaystyle b=\) the \(\displaystyle y\) intercept. 

Thus, the line that passes through the \(\displaystyle y\) axis at \(\displaystyle (0,-7)\) is \(\displaystyle y=\frac{3}{4}x-7\)

To find which equation of a line has the steepest slope, apply the formula: \(\displaystyle y=mx+b\), where \(\displaystyle m\) represents the slope of the line and \(\displaystyle b\) represents the \(\displaystyle y\) intercept.

Also, note that \(\displaystyle m=\frac{rise}{run}\), meaning the change in the \(\displaystyle y\) value, over the change in the \(\displaystyle x\) value. 

The equation that has the largest absolute value of m is the equation that has the steepest slope.

Thus, the equation \(\displaystyle y=-\frac{8}{3}x+4\) has the steepest slope, because in order to go from one point to the next move a vertical distance of \(\displaystyle 8\) and a horizontal distance of \(\displaystyle 3\) which is larger than any of the other choices. 

To find the slope of the line that passes through these two coordinate points, apply the formula: 

\(\displaystyle Slope=\frac{y_2-y_1}{x_2-x_1}\)

Thus the correct answer is:

\(\displaystyle \frac{ -1-4}{2--3}=\frac{-5}{5}=-\frac{5}{5}=-1\)

To find the slope of the line that passes through these two coordinate points, apply the formula: 

\(\displaystyle Slope=\frac{y_2-y_1}{x_2-x_1}\)

Thus the correct answer is:
\(\displaystyle \frac{6--3}{7--6}=\frac{6+3}{7+6}=\frac{9}{13}\)

To find the length of this line segment find the difference between each of the two end points \(\displaystyle x\) values, since they have the same \(\displaystyle y\) value. 

The difference between \(\displaystyle -8\) and \(\displaystyle 9\) is \(\displaystyle 17\).

\(\displaystyle 9-(-8)=9+8=17\)