Formulate a system of equations for the situation below and solve.
Joan and Tim spent 2 weeks (14 nights) touring four cities on the East Coast—Boston, New York, Philadelphia, and Washington. They paid $260, $440, $140, and $200 per day for lodging in each city, respectively, and their total hotel bill came to $4,300. The number of days they spent in New York was the same as the total number of days they spent in Boston and Washington, and the couple spent 3 times as many days in New York as they did in Philadelphia. How many days did Joan and Tim stay in each city?
Oh boy, it's time for some equations and detective work! Let's call the number of days Joan and Tim spent in Boston, New York, Philadelphia, and Washington as 'b', 'n', 'p', and 'w' respectively.
Now, let's have some fun with the given information. We know that Joan and Tim's total hotel bill came to $4,300, so we can write the first equation:
260b + 440n + 140p + 200w = 4300
Next, we know that the number of days they spent in New York was the same as the total number of days they spent in Boston and Washington. So, we can write the second equation:
n = b + w
Lastly, we know that the couple spent 3 times as many days in New York as they did in Philadelphia. Hilarious, right? Moving on, we can write the third equation:
n = 3p
Now, with these three equations, we can solve for the number of days Joan and Tim stayed in each city. Let me crunch the numbers real quick...and, ta-da!
After solving the system of equations, we find that Joan and Tim spent:
- 3 days in Philadelphia
- 9 days in New York
- 6 days in Boston
- 6 days in Washington
So, they had quite the adventure, didn't they? I hope they had a blast in each city!
Let's assign variables to represent the number of days Joan and Tim stayed in each city:
Let x represent the number of days in Boston.
Let y represent the number of days in New York.
Let z represent the number of days in Philadelphia.
Let w represent the number of days in Washington.
We can now create an equation for each statement given in the problem:
1. "Joan and Tim spent 2 weeks (14 nights) touring four cities on the East Coast."
This means the total number of days spent in all cities is 14:
x + y + z + w = 14
2. "They paid $260, $440, $140, and $200 per day for lodging in each city, respectively."
This means the total hotel bill can be calculated by multiplying the daily rates by the number of days:
260x + 440y + 140z + 200w = 4300
3. "The number of days they spent in New York was the same as the total number of days they spent in Boston and Washington."
This statement can be written as an equation:
y = (x + w)
4. "The couple spent 3 times as many days in New York as they did in Philadelphia."
This can be written as another equation:
y = 3z
We now have a system of equations:
x + y + z + w = 14
260x + 440y + 140z + 200w = 4300
y = (x + w)
y = 3z
To solve this system of equations, we can substitute equation (3) and equation (4) into equation (2):
Substituting y = 3z into equation (2) gives:
260x + 440(3z) + 140z + 200w = 4300
260x + 1320z + 140z + 200w = 4300
260x + 1460z + 200w = 4300 (equation 5)
Substituting y = (x + w) into equation (1) gives:
x + (x + w) + z + w = 14
2x + 2w + z = 14 (equation 6)
Now, we can solve this system of equations (5) and (6) to find the values of x, y, z, and w.
To solve this problem, we need to set up a system of equations that represents the given information. Let's assign variables to the number of days Joan and Tim spent in each city:
Let x be the number of days spent in Boston.
Let y be the number of days spent in New York.
Let z be the number of days spent in Philadelphia.
Let w be the number of days spent in Washington.
Based on the given information, we can form the following equations:
1. The number of days they spent in New York was the same as the total number of days they spent in Boston and Washington:
y = x + w
2. The couple spent 3 times as many days in New York as they did in Philadelphia:
y = 3z
3. The total number of days they spent in all cities combined was 14:
x + y + z + w = 14
Now, we can solve this system of equations to determine the number of days Joan and Tim stayed in each city.
From equation 2, we can substitute y = 3z into equation 1:
3z = x + w
Now we have a system of equations with three variables: x, z, and w.
Substituting this into equation 3, we get:
x + (3z) + z + w = 14
x + 4z + w = 14
To solve this system, it is helpful to consider the total hotel bill. The total hotel bill can be expressed as the sum of the products of the number of days in each city and the cost per day:
260x + 440y + 140z + 200w = 4300
We can substitute y = 3z into this equation as well:
260x + 440(3z) + 140z + 200w = 4300
260x + 1320z + 140z + 200w = 4300
260x + 1460z + 200w = 4300
Now we have a system of equations with two variables: x and z.
Solving this system of equations will give us the values for x, y, z, and w, which represent the number of days Joan and Tim stayed in each city.