Which is an equation of the given line in standard form? a. –8x + 9y = 23 b. –8x + 9y = –23 c. –8x + 7y = 25 d. –9x + 8y = –23

We have to calculate the slope of this line;
Given two points  (x₁,y₁) and (x₂,y₂) the slope of the line passes through these points is: 

m=(y₂-y₁)/(x₂-x₁)

We have many points in this line, we can choose any two points in this line. 
For example:
(-4,-1)
(1/2,3)
The slope will be:    m=(3+1)/(1/2+4)=4/(9/2)=8/9

Point-slope form of a line; We need a point (x₀,y₀)(any point of this line) and the slope "m";

y-y₀=m(x-x₀)

For example; the point can be ((-4,-1) and the slope of this line is 8/9
y+1=(8/9)(x+4)
y+1=8/9 x+32/9
9y/9+9/9=8/9 x+32/9
9y+9=8x+32
-8x+9y=32-9
-8x+9y=23  (equation in standard form)

A.  -8x+9y=23

Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.