Question

50 POINTS

Find an equation of the line that has slope -8 and passes through the midpoint of the line segment joining the points P1(-4,-3) and
P2(5,7)
Let x be the independent variable and y be the dependent variable. Please explain with steps!

Answer 1

Answer:

y = - 8x + 6

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given slope m = - 8 , then

y = - 8x + c ← is the partial equation

the midpoint of P1 (- 4, - 3 ) and P2 (5, 7 ) is the average of the x and y coordinates of P1 and P2 , that is

midpoint = ( $\frac{-4+5}{2}$ , $\frac{-3+7}{2}$ ) = ( $\frac{1}{2}$ , $\frac{4}{2}$ ) = ( $\frac{1}{2}$ , 2 )

Now

y = - 8x + c ← is the partial equation

to find c, substitute ( $\frac{1}{2}$ , 2 ) for x and y in the partial equation

2 = - 8 ($\frac{1}{2}$ ) + c = - 4 + c ( add 4 to both sides )

6 = c

y = - 8x + 6 ← equation of line

Answer 2

Answer:

$y=-8x+6$

Step-by-step explanation:

To find the equation of the line, we first need to find the coordinates of the midpoint of the line segment joining points P1(-4, -3) and P2(5, 7).

To find the midpoint, we can use the midpoint formula:

$\boxed{\begin{array}{l}\underline{\sf Midpoint \;formula}\\\\M(x,y) =\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)\\\\\textsf{where (x_1,y_1) and (x_2,y_2) are the endpoints.}\\\end{array}}$

Substitute the coordinates of the two points into the midpoint formula:

$M =\left(\dfrac{5+(-4)}{2},\dfrac{7+(-3)}{2}\right)$

$M =\left(\dfrac{5-4}{2},\dfrac{7-3}{2}\right)$

$M =\left(\dfrac{1}{2},\dfrac{4}{2}\right)$

$M =\left(\dfrac{1}{2},2\right)$

Therefore, the coordinates of the midpoint of the line segment joining P1(-4, -3) and P2(5, 7) are (1/2, 2).

Now that we have the slope of the line (m = -8) and a point on the line (1/2, 2), we can substitute both into the point-slope form of a linear equation:

$\begin{aligned}y-y_1&=m(x-x_1)\\\\y-2&=-8\left(x-\dfrac{1}{2}\right)\end{aligned}$

Simplify and rearrange to isolate y:

$\begin{aligned}y-2&=-8x+4\\\\y&=-8x+6\end{aligned}$

Therefore, the equation of a line that has slope -8 and passes through the midpoint of the line segment joining the points P1(-4, -3) and P2(5, 7) is:

$\Large\boxed{\boxed{y=-8x+6}}$