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)

Question

Responses

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=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+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=500s+75; Greenhill Country Club: C=1,000s+65

Answer 1

The correct pair of linear equations that represent the cost (C) for each location based on the number of students (s) is:

Regal Hotel: \( C = 75s + 500 \)

Greenhill Country Club: \( C = 65s + 1,000 \)

So the answer is:

Regal Hotel: \( C = 75s + 500 \); Greenhill Country Club: \( C = 65s + 1,000 \)

Answer 2

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; 10b+5c=12,500
b plus c equals 2,000 ; 10 b plus 5 c equals 12,500

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=12,500; 10b+5c=2,000

Answer 3

The situation can be modeled with the following pair of linear equations:

1. The total number of items (bags and caps):
   \( b + c = 2,000 \)

2. The total cost of the bags and caps within the budget:
   \( 10b + 5c = 12,500 \)

So the correct answer is:
b + c = 2,000; 10b + 5c = 12,500
b plus c equals 2,000; 10 b plus 5 c equals 12,500

Answer 4

Question
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)
Responses

Balloon Mania; $5
Balloon Mania; $5

Balloon Mania; $25 
Balloon Mania; $25 

Balloon Express; $25
Balloon Express; $25

Balloon Express; $5

Answer 5

Let's set up the equations for both companies based on the information provided.

For Balloon Express:

• Cost per balloon: $2
• Delivery cost: $10

The total cost for ordering 30 balloons from Balloon Express (let \( b \) be the number of balloons): \[ C_{Express} = 2b + 10 \] \[ C_{Express} = 2(30) + 10 = 60 + 10 = 70 \]

For Balloon Mania:

• Cost per balloon: $1.50
• Delivery cost: $20

The total cost for ordering 30 balloons from Balloon Mania: \[ C_{Mania} = 1.5b + 20 \] \[ C_{Mania} = 1.5(30) + 20 = 45 + 20 = 65 \]

Now, let's compare the costs:

• Total cost with Balloon Express: $70
• Total cost with Balloon Mania: $65

To find out how much they will save by choosing Balloon Mania: \[ \text{Savings} = C_{Express} - C_{Mania} = 70 - 65 = 5 \]

Therefore, the O’Donnells should choose Balloon Mania and they will save $5.

So the correct response is: Balloon Mania; $5

Answer 6

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)
Responses

12 quiches
12 quiches

75 quiches
75 quiches

20 quiches
20 quiches

45 quiches

Answer 7

To determine how many quiches the Spanish Club needs to sell to break even, we can set up an equation for their costs and revenue.

Cost to make the quiches:

• Fixed cost: $30 (labor)
• Variable cost per quiche: $0.50
• Total cost for \( q \) quiches: \[ \text{Total Cost} = 30 + 0.50q \]

Revenue from selling the quiches:

• Sale price per quiche: $2
• Total revenue from selling \( q \) quiches: \[ \text{Total Revenue} = 2q \]

To break even, the Total Revenue must equal the Total Cost: \[ 2q = 30 + 0.50q \]

Now, let's solve the equation:

1. Subtract \( 0.50q \) from both sides: \[ 2q - 0.50q = 30 \] \[ 1.5q = 30 \]

2. Divide both sides by 1.5: \[ q = \frac{30}{1.5} \] \[ q = 20 \]

Therefore, the Spanish Club will need to sell 20 quiches to break even.

So the correct response is: 20 quiches

Answer 8

You are trying to guess the code to a puzzle while participating in an escape room. There are two buttons, blue and red, and you have to put the correct number in each one to unlock the door to the next room. There are only two clues to help you:

Clue 1: If you double the blue number and add it to the red number, you get 6.
Clue 2: If you quadruple the blue number and add it to double the red number, you get 12.
You only get two attempts at the puzzle, and if you get it wrong, the game will be over. Use the system of equations shown to determine how many solutions there are to this puzzle.

{2b+r=64b+2r=12

(1 point)
Responses

There is no solution.
There is no solution.

There are only two solutions.
There are only two solutions.

There are infinitely many solutions.
There are infinitely many solutions.

There is only one solution.

Answer 9

Let's analyze the clues given to form a system of equations based on the variables:

Let:

• \( b \) = the blue number
• \( r \) = the red number

From Clue 1:

• If you double the blue number and add it to the red number, you get 6: \[ 2b + r = 6 \quad \text{(Equation 1)} \]

From Clue 2:

• If you quadruple the blue number and add it to double the red number, you get 12: \[ 4b + 2r = 12 \quad \text{(Equation 2)} \]

Now we can simplify Equation 2 by dividing everything by 2: \[ 2b + r = 6 \quad \text{(Equation 1)} \] \[ 2b + r = 6 \quad \text{(simplified Equation 2)} \]

Notice that both equations are identical. When you have two identical equations, it means that they represent the same line on a graph.

Solution Analysis:

1. Identical Equations: Since both equations are the same, there are infinitely many solutions, as any point on that line satisfies both equations.

Thus, the answer is: There are infinitely many solutions.