The Key Club is hosting a banquet as a fundraiser. If 60 people attend, they make $500. If 80 people attend, they make $1500.
Write an equation in slope-intercept form to describe the situation.
We can solve this problem by finding the slope-intercept form of the equation, which is \( y = mx + b \), where \( m \) is the slope, \( b \) is the y-intercept, \( y \) is the total money made, and \( x \) is the number of people attending.
First, let's determine the slope (m), which is the change in the amount made per additional attendee:
If attendance increases from 60 to 80 people, (which is an increase of 20 people), the money made increases by $1500 - $500 = $1000.
So, the slope \( m \) (money earned per additional person) is:
\[ m = \frac{1000}{20} = 50 \]
This means for each additional person who attends, the club makes $50.
Next, we need to find the y-intercept (b), which is the amount the club makes when no people attend (x = 0). We can use any of the given points to find this. Let's use the first point (x = 60, y = $500). Using the slope we've found:
\[ y = mx + b \]
\[ 500 = 50 \cdot 60 + b \]
\[ 500 = 3000 + b \]
\[ b = 500 - 3000 \]
\[ b = -2500 \]
This means if no one attends, the club would be "in the hole" $2500, which can represent fixed costs that are offset as more people attend.
Now we can write the equation in slope-intercept form:
\[ y = 50x - 2500 \]
This equation describes the club's earnings (y) based on the number of people attending (x), where 50 represents the additional money made per person, and -2500 represents the starting deficit or fixed costs.