Solve the problem.
-A vendor has learned that, by pricing hot dogs at $1.75, sales will reach 105 hot dogs per day. Raising the price to $2.75 will cause the sales to fall to 65 hot dogs per day. Let y be the number of hot dogs the vendor sells at x
Dollars each. Write a linear equation that models the number of hot dogs sold per day when the price is x
Dollars each.
A) y = 40x + 35
C) y = -40x - 175
D) y = -40x + 175

Question

Solve the problem.
-A vendor has learned that, by pricing hot dogs at $1.75, sales will reach 105 hot dogs per day. Raising the price to $2.75 will cause the sales to fall to 65 hot dogs per day. Let y be the number of hot dogs the vendor sells at x
Dollars each. Write a linear equation that models the number of hot dogs sold per day when the price is x
Dollars each.
A) y = 40x + 35 
C) y = -40x - 175 
D) y = -40x + 175

Quizplus · Accepted Answer

We can use the point-slope form of the equation to solve for the slope, which is the rate of change between the two given price and sales values:
slope = (change in y)/(change in x) = (65 - 105)/($2.75 - $1.75) = -40
Then, we can use one of the given points (either $1.75, 105) or ($2.75, 65) to solve for the y-intercept:
y - 105 = -40(x - 1.75)
y - 105 = -40x + 70
y = -40x + 175
Therefore, the linear equation that models the number of hot dogs sold per day when the price is x dollars each is y = -40x + 175, which matches choice D. We can use the slope-intercept form of a linear equation y = mx + b, where m is the slope and b is the y-intercept. The vendor sells 105 hot dogs at a price of $1.75, so we can use this point to find the slope:

105 = m(1.75) + b 
m = (105 - b) / 1.75

Similarly, using the point (65, $2.75), we get

65 = m(2.75) + b

Substituting the expression for m from the first equation into the second equation, we get

65 = [(105 - b) / 1.75] (2.75) + b 
65 = 6.25 - 2b/7 
b = 525/14

Substituting this expression for b into the first equation, we get

y = (-40/7) x + (525/14)

Simplifying, we get

y = -40x/7 + 75

Which is equivalent to answer choice D.

Solve the Problem. -A Vendor Has Learned That, by Pricing Hot Dogs at Hot