According to the point-slope form, the linear equation for the attendance in terms of the price is given as [tex]y=-20p+1490[/tex].
It is given to us that -
If the price of admission was $17, the attendance was about 1150 customers per day
When the price of admission was dropped to $12, attendance increased to about 1250 per day
We have to write a write a linear equation for the attendance in terms of the price.
Let us say that -
Price of admission is given as "p"
and, attendance of customers is given as "q"
It is given to u that -
For p = $17, q = 1150 customers per day
For p = $12, q = 1250 customers per day
We know that the point-slope form of a linear equation can be represented as -
[tex]y-y_{1} =m(x-x_{1} )[/tex] ------ (1)
where,
([tex]x_{1} ,y_{1}[/tex]) = initial coordinates of the point
m = slope of the linear equation = [tex]\frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex] ----- (2)
From the given information, we can represent the points as -
([tex]x_{1} ,y_{1}[/tex]) = (17, 1150)
and, ([tex]x_{2} ,y_{2}[/tex]) = (12, 1250)
Substituting the values of ([tex]x_{1} ,y_{1}[/tex]) and ([tex]x_{2} ,y_{2}[/tex]) in equation (2), we have -
[tex]m = \frac{y_{2} -y_{1} }{x_{2} -x_{1} }\\= > m = \frac{1250-1150}{12-17}\\= > m = \frac{100}{-5} \\= > m = -20[/tex]------ (3)
Now, substituting the values of ([tex]x_{1} ,y_{1}[/tex]) and m from equation (3) in the point-slope form of the linear equation (1), we have
[tex]y-y_{1} =m(x-x_{1} )\\= > y-1150=(-20)(x-17)\\= > y-1150=-20x+340\\= > y=-20x+340+1150\\= > y=-20x+1490[/tex]
In terms of price, p we can write the above linear equation as -
[tex]y=-20p+1490[/tex]
Thus, the linear equation for the attendance in terms of the price is given as [tex]y=-20p+1490[/tex].
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