The drama club is selling tickets to their play to raise money for the show's expenses. Each student ticket sells for $5.50 and each adult ticket sells for $9. The auditorium can hold at most 140 people. The drama club must make a minimum of $990 from ticket sales to cover the show's costs. If xx represents the number of student tickets sold and yy represents the number of adult tickets sold, write and solve a system of inequalities graphically and determine one possible solution.
To write the system of inequalities, we need to consider two things:
1) The auditorium can hold at most 140 people: xx + yy ≤ 140
2) The drama club must make a minimum of $990: 5.50xx + 9yy ≥ 990
To graphically solve the system of inequalities, we need to plot the boundary lines for each inequality and shade the feasible region. Let's start with the first inequality:
The boundary line for xx + yy ≤ 140 is xx + yy = 140. This line represents the maximum number of people that can be in the auditorium. To graph this line, we plot the points (0, 140) and (140, 0) and draw a line connecting them.
Next, let's move on to the second inequality:
The boundary line for 5.50xx + 9yy ≥ 990 can be rewritten as 11xx + 18yy ≥ 1980 by multiplying both sides by 2. To graph this line, we plot the points (0, 110) and (180, 0) and draw a line connecting them.
Now, we need to shade the feasible region. Since we want to find the number of student tickets (xx) and adult tickets (yy) sold, the feasible region must be in the first quadrant (positive values only).
Therefore, the feasible region is above the xx + yy = 140 line and above the 11xx + 18yy ≥ 1980 line.
By inspecting the graph, we can see that one possible solution is (40, 100). This means that 40 student tickets and 100 adult tickets were sold, which results in a total revenue of $5.50(40) + $9(100) = $550 + $900 = $1450.
please solve each inequality for yy
To solve the first inequality, we can rearrange it to solve for yy:
xx + yy ≤ 140
yy ≤ 140 - xx
Now, let's solve the second inequality for yy:
5.50xx + 9yy ≥ 990
Subtract 5.50xx from both sides:
9yy ≥ 990 - 5.50xx
Divide both sides by 9:
yy ≥ (990 - 5.50xx) / 9
So, the two inequalities are:
yy ≤ 140 - xx
yy ≥ (990 - 5.50xx) / 9
Now, we have a system of inequalities where yy has been isolated in terms of xx.
One possible solution using this system of inequalities is when xx = 40. In this case, the inequalities become:
yy ≤ 140 - 40
yy ≥ (990 - 5.50(40)) / 9
Simplifying:
yy ≤ 100
yy ≥ 100
So, one possible solution is xx = 40 and yy = 100, where the drama club sells 40 student tickets and 100 adult tickets.