Question
The Walker and Baldwin families ordered lunch at the refreshment stand at the basketball game. The Walker family ordered 4 hot dogs and 3 cheeseburgers and paid $23.75. The Baldwin family ordered 6 hots dogs and 2 cheeseburgers and paid $25. Write a system of equations using h
for hot dogs and c for cheeseburgers.
What are the prices of a hot dog and a cheeseburger?
This question requires you to show your work.
for hot dogs and c for cheeseburgers.
What are the prices of a hot dog and a cheeseburger?
This question requires you to show your work.
Answers
GPT-4o mini
To solve this problem, we'll start by defining variables for the prices of hot dogs and cheeseburgers. Let:
- \( h \) = price of one hot dog
- \( c \) = price of one cheeseburger
Next, we can set up a system of equations based on the information provided for both families.
### Setting Up the Equations
1. **Walker Family**:
- They ordered 4 hot dogs and 3 cheeseburgers for a total cost of $23.75.
- The equation can be written as:
\[
4h + 3c = 23.75
\]
2. **Baldwin Family**:
- They ordered 6 hot dogs and 2 cheeseburgers for a total cost of $25.00.
- The equation can be written as:
\[
6h + 2c = 25
\]
### System of Equations
Now we have the following system of equations:
\[
\begin{align*}
1) & \quad 4h + 3c = 23.75 \\
2) & \quad 6h + 2c = 25
\end{align*}
\]
### Solving the System of Equations
We can solve this system using the substitution or elimination method. Here, we'll use the elimination method.
First, let's multiply the second equation by \( 1.5 \) to make the coefficients of \( c \) the same:
\[
1.5(6h + 2c) = 1.5(25)
\]
\[
9h + 3c = 37.5
\]
Now our updated system of equations is:
\[
\begin{align*}
1) & \quad 4h + 3c = 23.75 \\
2) & \quad 9h + 3c = 37.5
\end{align*}
\]
### Subtracting the First Equation from the Second
Now, we can eliminate \( c \) by subtracting the first equation from the second:
\[
(9h + 3c) - (4h + 3c) = 37.5 - 23.75
\]
\[
5h = 13.75
\]
Dividing both sides by \( 5 \):
\[
h = \frac{13.75}{5} = 2.75
\]
### Finding the Price of Cheeseburgers \( c \)
Now that we have the value of \( h \), we can substitute it back into one of the original equations to find \( c \). We can use the first equation:
\[
4h + 3c = 23.75
\]
Substituting \( h = 2.75 \):
\[
4(2.75) + 3c = 23.75
\]
\[
11 + 3c = 23.75
\]
Subtracting 11 from both sides:
\[
3c = 23.75 - 11
\]
\[
3c = 12.75
\]
Dividing by 3:
\[
c = \frac{12.75}{3} = 4.25
\]
### Final Prices
Thus, the prices of the hot dog and cheeseburger are:
- Price of a hot dog (\( h \)): **$2.75**
- Price of a cheeseburger (\( c \)): **$4.25**
- \( h \) = price of one hot dog
- \( c \) = price of one cheeseburger
Next, we can set up a system of equations based on the information provided for both families.
### Setting Up the Equations
1. **Walker Family**:
- They ordered 4 hot dogs and 3 cheeseburgers for a total cost of $23.75.
- The equation can be written as:
\[
4h + 3c = 23.75
\]
2. **Baldwin Family**:
- They ordered 6 hot dogs and 2 cheeseburgers for a total cost of $25.00.
- The equation can be written as:
\[
6h + 2c = 25
\]
### System of Equations
Now we have the following system of equations:
\[
\begin{align*}
1) & \quad 4h + 3c = 23.75 \\
2) & \quad 6h + 2c = 25
\end{align*}
\]
### Solving the System of Equations
We can solve this system using the substitution or elimination method. Here, we'll use the elimination method.
First, let's multiply the second equation by \( 1.5 \) to make the coefficients of \( c \) the same:
\[
1.5(6h + 2c) = 1.5(25)
\]
\[
9h + 3c = 37.5
\]
Now our updated system of equations is:
\[
\begin{align*}
1) & \quad 4h + 3c = 23.75 \\
2) & \quad 9h + 3c = 37.5
\end{align*}
\]
### Subtracting the First Equation from the Second
Now, we can eliminate \( c \) by subtracting the first equation from the second:
\[
(9h + 3c) - (4h + 3c) = 37.5 - 23.75
\]
\[
5h = 13.75
\]
Dividing both sides by \( 5 \):
\[
h = \frac{13.75}{5} = 2.75
\]
### Finding the Price of Cheeseburgers \( c \)
Now that we have the value of \( h \), we can substitute it back into one of the original equations to find \( c \). We can use the first equation:
\[
4h + 3c = 23.75
\]
Substituting \( h = 2.75 \):
\[
4(2.75) + 3c = 23.75
\]
\[
11 + 3c = 23.75
\]
Subtracting 11 from both sides:
\[
3c = 23.75 - 11
\]
\[
3c = 12.75
\]
Dividing by 3:
\[
c = \frac{12.75}{3} = 4.25
\]
### Final Prices
Thus, the prices of the hot dog and cheeseburger are:
- Price of a hot dog (\( h \)): **$2.75**
- Price of a cheeseburger (\( c \)): **$4.25**