Step-by-step explanation:
the y-intercept is the y-value, when x = 0 (as x = 0 is the equation for the y-axis : all points for which x = 0).
in the usual slope-intercept form of a line equation
y = ax + b
b is the y-intercept. a is the slope, which is the ratio
y coordinate difference / x coordinate difference
when going from one point on the line to another.
so, when going from A to B we get
x changes by -5 (from 8 to 3).
y changes by +7 (from 0 to 7).
the slope a is therefore +7/-5 = -7/5
our equation is now
y = (-7/5)x + b
we use now one of the given points to get b :
0 = (-7/5)×8 + b = -56/5 + b
b = y-intercept of AB = 56/5
now a parallel lines goes through D (5, 5).
a parallel line has the same slope (otherwise it would not be parallel).
so, again
y = (-7/5)x + b
and we use the given point to get b :
5 = (-7/5)×5 + b = -7 + b
b = 5 + 7 = 12
and the equation for CD is
y = (-7/5)x + 12
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]