Question

At a certain food truck, hot dogs cost $3
and burgers cost $5
. A customer ordered 7
items and spent $27
.

) If x
is the number of burgers the customer bought and y
is the number of hot dogs, write a system of equations that models the problem and graph it.

) How many of each item were purchased?

Answer 1

To model the problem using a system of equations, we have two types of items (burgers and hot dogs) and two constraints given by the problem (the total number of items and the total cost).

Let:
- \( x \) be the number of burgers
- \( y \) be the number of hot dogs

From the problem, we can form the following equations:

1. The total number of items equation:
\[
x + y = 7
\]

2. The total cost equation:
\[
5x + 3y = 27
\]

Now, we have the system of equations:
\[
\begin{align*}
1. & \quad x + y = 7 \quad \text{(1)} \\
2. & \quad 5x + 3y = 27 \quad \text{(2)}
\end{align*}
\]

### Solving the System of Equations

We can solve this system using substitution or elimination. Let's use substitution by solving equation (1) for \( y \):

\[
y = 7 - x
\]

Now, we can substitute \( y \) in equation (2):

\[
5x + 3(7 - x) = 27
\]

Expanding and simplifying:

\[
5x + 21 - 3x = 27
\]

\[
2x + 21 = 27
\]

Subtracting 21 from both sides:

\[
2x = 6
\]

Dividing by 2:

\[
x = 3
\]

Now plug \( x \) back into equation (1) to find \( y \):

\[
3 + y = 7
\]

Subtracting 3 from both sides:

\[
y = 4
\]

### Conclusion

The customer purchased:
- 3 burgers
- 4 hot dogs

### Verification of the Solution

To verify, we can check both equations with \( x = 3 \) and \( y = 4 \):

1. \( x + y = 7 \):
\[
3 + 4 = 7 \quad \text{(True)}
\]

2. \( 5x + 3y = 27 \):
\[
5(3) + 3(4) = 15 + 12 = 27 \quad \text{(True)}
\]

Both equations are satisfied, confirming that the solutions are correct.