Real-World Systems of Equations Quick Check
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Question
The senior class at Hartford High School is planning their prom. They’ve narrowed the location down to two choices. The Regal Hotel charges, C, $500 for the ballroom rental fee and $75 per student, s, for the meal. The Greenhill Country Club charges, C, $1,000 for the ballroom rental fee and $65 per student, s, for the meal. Create a pair of linear equations that represent this real-world problem.(1 point)
Responses
Regal Hotel: C=75s+1,000; Greenhill Country Club: C=65s+500
, Regal Hotel: , upper C equals 75 s plus 1,000, ; Greenhill Country Club: , upper C equals 65 s plus 500
Regal Hotel: C=1,000s+75; Greenhill Country Club: C=500s+65
, Regal Hotel: , upper C equals 1,000 s plus 75, ; Greenhill Country Club: , upper C equals 500 s plus 65
Regal Hotel: C=75s+500; Greenhill Country Club: C=65s+1,000
Regal Hotel: , upper C equals 75 s plus 500, ; Greenhill Country Club: , upper C equals 65 s plus 1,000
Regal Hotel: C=500s+75; Greenhill Country Club: C=1,000s+65
, Regal Hotel: , upper C equals 500 s plus 75, ; Greenhill Country Club: , upper C equals 1,000 s plus 65
The Jackson Jammers are giving away bags and caps at their next game. The bags, b, cost them $10 each, and the caps, c, cost them $5 each. They plan to give away 2,000 total items and have a budget of $12,500. Write the pair of linear equations that model this situation.(1 point)
Responses
b+c=2,000; 5b+10c=12,500
, b plus c equals 2,000, ; , 10 b plus 10 c equals 12,500
b+c=12,500; 5b+10c=2,000
, b plus c equals 12,500, ; , 5 b plus 10 c equals 2,000
b+c=2,000; 10b+5c=12,500
b plus c equals 2,000, ; , 10 b plus 5 c equals 12,500
b+c=12,500; 10b+5c=2,000
, b plus c equals 2,000, ; , 10 b plus 5 c equals 2,000
Solve this real-world problem using a system of equations. The O’Donnells are having a birthday party for their son, and they plan to have a clown bring balloons to the party. Balloon Express charges $2 per balloon and $10 for delivery. Balloon Mania charges $1.50 per balloon and $20 for delivery. If the O’Donnells plan to order 30 balloons, which company should they choose, and how much will they save?(1 point)
Let's use the following variables:
- Let B represent the number of balloons ordered
- Let C represent the cost of the order, including delivery
Using this information, we can create a system of equations:
- Balloon Express: C = 2B + 10
- Balloon Mania: C = 1.5B + 20
We know that the O’Donnells plan to order 30 balloons, so we can substitute B = 30 in the equations:
- Balloon Express: C = 2(30) + 10 = 70
- Balloon Mania: C = 1.5(30) + 20 = 65
Therefore, the O'Donnells should choose Balloon Mania because they will save $5 compared to Balloon Express.
The Spanish Club at Santa Fe Middle School wants to sell mini quiches to raise money for a field trip. The culinary department at their school will make the quiches for $30 plus $0.50 per quiche for the ingredients. The Spanish Club wants to sell each quiche for $2. How many quiches will they need to sell in order to break even?(1 point)
Let's use the following variables:
- Let Q represent the number of quiches sold
- Let C represent the cost of making the quiches
Using this information, we can create an equation for the total cost to make the quiches:
- C = 30 + 0.5Q
We also know that the Spanish Club will sell each quiche for $2, so the total revenue from selling Q quiches will be:
- R = 2Q
In order to break even, the total revenue from selling the quiches must equal the total cost of making the quiches. Therefore, we can set R equal to C and solve for Q:
- 2Q = 30 + 0.5Q
- 1.5Q = 30
- Q = 20
Therefore, the Spanish Club will need to sell 20 mini quiches in order to break even.
The Baynard High School Boosters are selling hot dogs and pretzels at the concession stand. They have 300 hot dogs and pretzels in stock and want to earn an income of $800. They plan to sell the hot dogs for $4 each and the pretzels for $2 each. How many hot dogs and how many pretzels do the boosters need to sell to reach their goal of $800?(1 point)
Responses
50 hot dogs and 250 pretzels
50 hot dogs and 250 pretzels
200 hot dogs and 100 pretzels
200 hot dogs and 100 pretzels
250 hot dogs and 50 pretzels
250 hot dogs and 50 pretzels
100 hot dogs and 200 pretzels
Let's use the following variables:
- Let H represent the number of hot dogs sold
- Let P represent the number of pretzels sold
Using this information, we can create a system of equations:
- Income: 4H + 2P = 800
- Quantity: H + P = 300
We can use the Quantity equation to solve for one of the variables in terms of the other:
- P = 300 - H
Substitute this expression for P into the Income equation and solve for H:
- 4H + 2(300 - H) = 800
- 4H + 600 - 2H = 800
- 2H = 200
- H = 100
Therefore, the Boosters need to sell 100 hot dogs to earn $400. To make up the remaining $400, they must sell 200 pretzels, which they can check using either equation in the system:
- H + P = 300
- 100 + P = 300
- P = 200
Therefore, the Boosters need to sell 100 hot dogs and 200 pretzels to reach their goal of $800.
To create a pair of linear equations for this problem, we need to express the total cost (C) for each location in terms of the number of students (s) attending the prom.
For the Regal Hotel, the ballroom rental fee is $500, so the equation for the total cost at the Regal Hotel is C = 500 + 75s.
For the Greenhill Country Club, the ballroom rental fee is $1,000, so the equation for the total cost at the Greenhill Country Club is C = 1,000 + 65s.
Therefore, the correct pair of linear equations that represent this real-world problem is:
Regal Hotel: C = 500 + 75s
Greenhill Country Club: C = 1,000 + 65s
So the correct answer is:
Regal Hotel: C = 500 + 75s; Greenhill Country Club: C = 1,000 + 65s