Student tickets to the Homecoming game cost $5 each. General admission tickets cost $8 each. So far, 150 tickets have been sold. $900 has been collected.
A. Write a system of equations for this model in standard form.
B. Graph the system. ( let each square represents 20 units)
C. What is the solution to the system?
D. Is (180, -30) a solution? Explain.
Step 1:Identify target unknowns.
Identify them as variables.
Recall that variables do not have to be X and Y. They could be initials to the target unknowns, such as C for carrots, and P for potatoes.
Here we have
G=number of general admissions sold
S=number of student tickets sold
Step 2:
Express the relevant facts as mathematical equations.
"150 tickets have been sold" =>
S+G=150...............(1)
"$900 has been collected"
means total revenue is $900.
Revenue from students=5S
Revenue from general=8G
So
5S+8G=900................(2)
Step 3:Solve the equations set up in step 2.
S+G=150 ...............(1)
5S+8G=900 ................(2)
Here we solve by first eliminating S
(2)-5(1)
5S+8G - (5S+5G) = 900-5(150)
simplify
5S-5S+8G-5G=900-750 =>
3G=150
G=50
So 50 General admission tickets were sold.
Substitute G in equation (1) to find S, the number of student tickets sold is your next step.
Step 4:
Once you have found the solutions, substitute back into equations (1) and (2) to check for correctness of the answers.
Above solves parts (A) and (C).
To do graphing, you need a good video or class for that. Try for example:
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-solutions-to-two-var-linear-equations/v/graphs-of-linear-equations
or use the graphing tool Desmos at
Desmos.com to help you.
Part D:
Propose and post an answer, we can discuss.
A. To write a system of equations for this model in standard form, let's define two variables:
Let x represent the number of student tickets sold.
Let y represent the number of general admission tickets sold.
The cost of a student ticket is $5, so the revenue from selling student tickets can be represented as 5x.
Similarly, the cost of a general admission ticket is $8, so the revenue from selling general admission tickets can be represented as 8y.
The total number of tickets sold is given as 150, so we can write the first equation:
x + y = 150
The total amount collected from selling tickets is $900, so we can write the second equation:
5x + 8y = 900
B. To graph the system, we can convert each equation to slope-intercept form (y = mx + b):
Equation 1:
x + y = 150
y = -x + 150
Equation 2:
5x + 8y = 900
8y = -5x + 900
y = (-5/8)x + 112.5
Now, let's plot the graph:
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C. To find the solution to the system, we need to find the point of intersection between the two lines on the graph. The point of intersection represents the number of student tickets (x) and general admission tickets (y) that satisfy both equations.
By looking at the graph, we can estimate that the point of intersection is approximately (60,90). Therefore, the solution to the system is x = 60 and y = 90.
D. To check if (180, -30) is a solution, we substitute the values into both equations:
Equation 1:
180 + (-30) = 150
150 = 150
The first equation is satisfied.
Equation 2:
5(180) + 8(-30) = 900
900 = 900
The second equation is also satisfied.
Therefore, (180, -30) is a valid solution to the system of equations.