Answer:
[tex]\textsf{$y$-intercept of $\overleftrightarrow{AB}$}:\quad \dfrac{56}{5}[/tex]
[tex]\textsf{Equation of line $\overleftrightarrow{CD}$}:\quad y=-\dfrac{7}{5}x+12[/tex]
Step-by-step explanation:
To find the y-intercept of line AB, we need to determine the equation of the line.
First find the slope of the line by substituting the coordinates of the given points on the line A(8, 0) and B(3, 7) into the slope formula:
[tex]\textsf{Slope}\:(m)=\dfrac{y_B-y_A}{x_B-x_A}=\dfrac{7-0}{3-8}=-\dfrac{7}{5}[/tex]
Now, substitute the slope m = -7/5 and one of the points (8, 0) into the point-slope formula:
[tex]y-y_1=m(x-x_1)\\\\\\y-0=-\dfrac{7}{5}(x-8)\\\\\\y=-\dfrac{7}{5}(x-8)[/tex]
Expand the brackets so that the equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept:
[tex]y=-\dfrac{7}{5}x+\dfrac{56}{5}[/tex]
Therefore, the y-intercept of line AB is:
[tex]\Large\boxed{\boxed{\dfrac{56}{5}}}[/tex]
As line CD is parallel to line AB, the equations of their lines will have the same slope, m = -7/5.
To find the equation of line CD, substitute the slope m = -7/5 and the coordinates of point D (5, 5) into the the point-slope formula:
[tex]y-y_1=m(x-x_1)\\\\\\y-5=-\dfrac{7}{5}(x-5)\\\\\\y-5=-\dfrac{7}{5}x+7\\\\\\y=-\dfrac{7}{5}x+12[/tex]
Therefore, the equation of line CD is:
[tex]\Large\boxed{\boxed{y=-\dfrac{7}{5}x+12}}[/tex]
