Hudson is blocking off several rooms in a hotel for guests coming to his wedding. The hotel can reserve small rooms that can hold 3 people, and large rooms that can hold 4 people.... Hudson is blocking off several rooms in a hotel for guests coming to his wedding. The hotel can reserve small rooms that can hold 3 people, and large rooms that can hold 4 people. Hudson reserved twice as many large rooms as small rooms, which altogether can accommodate 66 guests. Write a system of equations that could be used to determine the number of small rooms reserved and the number of large rooms reserved. Define the variables that you use to write the system.
Understand the Problem
The question is asking us to formulate a system of equations based on the information provided about small and large rooms reserved for a wedding. The task involves defining variables for the number of each type of room and using the constraints given (total guests and relationship between room types) to create the equations.
Answer
The system is: $3s + 4l = 66$, $l = 2s$, with $s = 6$ and $l = 12$.
Answer for screen readers
The system of equations that could be used is:
- $3s + 4l = 66$
- $l = 2s$
Where ( s = 6 ) (number of small rooms) and ( l = 12 ) (number of large rooms).
Steps to Solve
- Define Variables
Let:
- $s$ = number of small rooms reserved
- $l$ = number of large rooms reserved
- Set Up the First Equation
Each small room accommodates 3 guests, and each large room accommodates 4 guests. The total number of guests to be accommodated is 66. Thus, the first equation can be expressed as: $$ 3s + 4l = 66 $$
- Set Up the Second Equation
According to the problem, Hudson reserved twice as many large rooms as small rooms. This relationship gives us the second equation: $$ l = 2s $$
- Combine the Equations
Now we have two equations:
- ( 3s + 4l = 66 )
- ( l = 2s )
We can substitute the second equation into the first equation to solve for $s$.
- Substituting the Second Equation into the First
Replace $l$ in the first equation: $$ 3s + 4(2s) = 66 $$ This simplifies to: $$ 3s + 8s = 66 $$
- Solve for $s$
Combine like terms: $$ 11s = 66 $$
Now, divide both sides by 11: $$ s = 6 $$
- Find the Value of $l$
Using the value of $s$ in the second equation: $$ l = 2s = 2(6) = 12 $$
The system of equations that could be used is:
- $3s + 4l = 66$
- $l = 2s$
Where ( s = 6 ) (number of small rooms) and ( l = 12 ) (number of large rooms).
More Information
In this problem, we've created a system of equations based on the information provided. The small rooms hold 3 guests, and the large rooms hold 4 guests. This method of defining variables and setting up equations is commonly used in algebra to solve real-world problems.
Tips
- Confusing the number of guests that each type of room can hold.
- Misinterpreting the relationship between the number of small and large rooms. Double-checking that "twice as many" is correctly represented in the equations is crucial.
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