RESTAURANTS In 2012, a popular pizza franchise had 2483 restaurants. In 2017, there were 2606 franchised restaurants. Write an equation in point-slope form that represents the number of restaurants y that are franchised x years after 2010.
To represent the number of restaurants y that are franchised x years after 2010, we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Given that in 2012, the number of restaurants was 2483 and in 2017, the number of restaurants was 2606, we have the points (2,483) and (7,606).
Using the point-slope form, we can choose one point and calculate the slope:
m = (y2 - y1) / (x2 - x1)
m = (2606 - 2483) / (7 - 2) = 123 / 5 = 24.6
Now, substitute the values from one of the points into the equation:
y - y1 = m(x - x1)
y - 2483 = 24.6(x - 2)
Thus, the equation in point-slope form that represents the number of restaurants y that are franchised x years after 2010 is:
y - 2483 = 24.6(x - 2)
To write the equation in point-slope form, we need to determine the slope and a point on the line.
The slope represents the rate of change in the number of restaurants per year. We can calculate the slope (m) using the formula:
m = (change in y) / (change in x)
In this case, the "change in y" is the difference in the number of restaurants between 2017 and 2012: 2606 - 2483 = 123.
The "change in x" is the difference in years between 2017 and 2012: 2017 - 2012 = 5.
So, our slope (m) is:
m = 123 / 5 = 24.6
Now, let's find a point on the line. Since the question asks for the number of restaurants x years after 2010, we can use (0, 2483) as the point. This represents the number of restaurants in 2012 (zero years after 2010).
Finally, we can write the equation in point-slope form:
y - y1 = m(x - x1)
Plugging in the values:
y - 2483 = 24.6(x - 0)
Simplifying:
y - 2483 = 24.6x
This is the equation in point-slope form that represents the number of franchised restaurants y x years after 2010.